CHAPTER OBJECTIVES Triangle Trigonometry

Transcription

1 9 Tringle Trigonometry Trcts of lnd re often mde up of irregulr shpes creted y geogrphicl fetures such s lkes nd rivers. Surveyors often mesure these irregulrly shped trcts of lnd y dividing them into tringles. To clculte the re, side lengths, nd ngle mesures of ech tringle, they must etend their knowledge of right tringle trigonometry to include tringles tht hve no right ngle. In this chpter you ll lern how to do these clcultions. CHAPTER OBJECTIVES Given two sides nd the included ngle of tringle, find y direct mesurement the length of the third side of the tringle. Given two sides nd the included ngle of tringle, derive nd use the lw of cosines to find the length of the third side. Given three sides of tringle, find n ngle mesure. Given the mesures of two sides nd the included ngle, find the re of the tringle. Given the mesure of n ngle, the length of the side opposite this ngle, nd one other piece of informtion out tringle, find the other side lengths nd ngle mesures. C Given two sides of tringle nd non-included ngle, clculte the possile lengths of the third side. Given two vectors, dd them to find the resultnt vector. Given rel-world prolem, identify tringle nd use the pproprite technique to clculte unknown side lengths nd ngle mesures. A c B

2 Chpter 9 Tringle Trigonometry Overview This chpter egins with the lw of cosines, first discovered y mesuring ccurtely drwn grphs nd then y proving it with lgeric methods. Students then lern the re formul hlf of side times side times sine of included ngle, which leds to the lw of sines. This re formul lso lys the foundtion for the cross product of vectors in Chpter 1. The miguous cse is pproched through single clcultion using the lw of cosines. Section 9-6, on vector ddition, cn e used to introduce students to the unit vectors in the - nd y-directions, lthough some instructors prefer to postpone vectors until Chpter 1. The chpter concludes with rel-world, tringle prolems where students must decide which tringle techniques to use. A cumultive review of Chpters through 9 ppers in Section 9-9. Using This Chpter This is the finl chpter in Unit : Trigonometric nd Periodic Functions. This chpter is n etension of the trigonometry students lerned in geometry nd erlier in this course. Consider spending some etr time on the cumultive review to mke sure students grsp trigonometric concepts which form uilding locks for clculus. Following this chpter, continue to Chpter 10, or, for those who studied trigonometry erly in the yer, return to Chpter. Teching Resources Eplortions Eplortion 9-1: Introduction to Olique Tringles Eplortion 9-: Derivtion of the Lw of Cosines Eplortion 9-: Angles y Lw of Cosines Eplortion 9-3: Are of Tringle nd Hero s Formul Eplortion 9-3: Derivtion of Hero s Formul Eplortion 9-4: The Lw of Sines Eplortion 9-4: The Lw of Sines for Angles Eplortion 9-: The Amiguous Cse, SSA Eplortion 9-: Golf Bll Prolem Eplortion 9-6: Sum of Two Displcement Vectors Eplortion 9-6: Nvigtion Vectors Eplortion 9-7: The Ship s Pth Prolem Eplortion 9-7: Are of Regulr Polygon Blckline Msters Sections 9-8 nd 9-9 Supplementry Prolems Sections 9-, 9-3, nd 9- to 9-8 Assessment Resources Test 4, Sections 9-1 to 9-4, Forms A nd B Test, Chpter 9, Forms A nd B Test 6, Cumultive Test, Chpters 9, Forms A nd B Technology Resources Dynmic Preclculus Eplortions Lw of Cosines Vrile Tringle Lw of Sines Sketchpd Presenttion Sketches Lw of Cosines Present.gsp Lw of Sines Present.gsp Activities Sketchpd: Tringles nd Squres: The Lw of Cosines Sketchpd: The Lw of Sines CAS Activity 9-: The Lw of Sines vs. the Lw of Cosines CAS Activity 9-: An Alterntive to the Lws of Sines nd Cosines Clcultor Progrms AREGPOLY 441A Chpter 9 Interlef: Tringle Trigonometry

3 Stndrd Schedule Pcing Guide Dy Section Suggested Assignment Introduction to Olique Tringles Olique Tringles: The Lw of Cosines RA, Q1 Q10, 1, 3, 6, 7 13 odd, 14, 1, 17, Are of Tringle RA, Q1 Q10, 1, 3, 7 9, 11, 13, Olique Tringles: The Lw of Sines RA, Q1 Q10, 1 9 odd, 10, 11, 13, 14 RA, Q1 Q10, 1 13 odd, The Amiguous Cse Quiz/test students on the mteril in Sections 9-, ssign selection 6 of prolems not previously ssigned, or use the dy to recp the chpter concepts 7 RA, Q1 Q10, 1, 3,, 7, Vector Addition , 17, 19, 1,, 4 9 RA, Q1 Q10, 1-9 odd 9-7 Rel-World Tringle Prolems odd, 18, nd hve students write their own prolems 11 R0 R7, T1 T1 9-8 Chpter Review nd Test 1 Cumultive Review Cumultive Review Cumultive Review, Chpters 9 14 Cumultive Review 38 46, Prolem Set 10-1 Block Schedule Pcing Guide Dy Section Suggested Assignment 1 9- Olique Tringles: The Lw of Cosines RA, Q1 Q10, 1, 3, 6, 7 13 odd 9-3 Are of Tringle RA, Q1 Q10, 1, 3, 7, Are of Tringle 9, 11, Olique Tringles: The Lw of Sines RA, Q1 Q10, 1 9 odd, 10, The Amiguous Cse RA, Q1 Q10, 1 13 odd, Vector Addition RA, Q1 Q10, 1, 3, 7, 9, 13 1 odd, 9-7 Rel-World Tringle Prolems RA, Q1 Q10, 1, 3, 6, 7, 9, 13, 1, Chpter Review R0 R7, T1 T Chpter Test 9-9 Cumultive Review, Chpters 9 Cumultive Review Cumultive Review, Chpters 9 Cumultive Review Qudrtic Reltions nd Conic Sections Prolem Set 10-1 Chpter 9 Interlef 441B

4 Section 9-1 Clss Time 1 dy PLANNING Homework Assignment Prolems 1 6 Teching Resources Eplortion 9-1: Introduction to Olique Tringles Technology Resources Eplortion 9-1: Introduction to Olique Tringles TEACHING Importnt Terms nd Concepts Olique tringle Section Notes Section 9-1 sets the stge for the development of the lw of cosines in Section 9-. You cn ssign this section for homework the night of the Chpter 8 test or s group ctivity to e completed in clss. No clssroom discussion is needed efore students egin the ctivity. You my wnt to dpt the prolem set so tht it cn e done with The Geometer s Sketchpd or Fthom. Eplortion Notes Eplortion 9-1 my e ssigned in plce of Eplortory Prolem Set 9-1. It covers much of the sme mteril ut includes grid for students to plot side s function of ngle A y hnd. Encourge students to plot the points with gret cre. The eplortion could lso e used s review sheet. Allow students 0 minutes to complete this ctivity. Mthemticl Overview 44 Chpter 9: Technology Notes GRAPHICALLY ALGEBRAICALLY NUMERICALLY VERBALLY The Pythgoren theorem descries how to find the length of the hypotenuse of right tringle if you know the lengths of the two legs. If the ngle formed y the two given sides is not right ngle, you cn use the lw of cosines (n etension of the Pythgoren theorem) to find side lengths or ngle mesures. If two ngles nd side opposite one of the ngles or two sides nd n ngle opposite one of the sides re given, you cn use the lw of sines. The re cn lso e clculted from side nd ngle mesures. These techniques give you wy to nlyze vectors, which re quntities (such s velocity) tht hve oth direction nd mgnitude. You will lern out these techniques in four wys. Mke scle drwing of the tringle using the given informtion, nd mesure the length of the third side. A c Lw of cosines: B Are of tringle: The length of side : The re of the tringle: Eplortion 9-1 in the Instructor s Resource Book hs students mesure the lengths of legs of vrious tringles nd then conjecture the lw of cosines. The dt they gther cn esily e recorded nd plotted in Fthom. C c c cos B A 1 c sin B (149)(37) cos ft A 1 (37)(149) sin 13 14, ,808 ft Given two sides of tringle nd the included ngle, the lw of cosines cn e used to find the length of the third side nd the sine cn e used to find the re. If ll three sides re given, the lw of cosines cn e used in reverse to find ny ngle mesure. 44 Chpter 9: Tringle Trigonometry

5 9-1 Ojective Eplortory Prolem Set 9-1 Introduction to Olique Tringles You lredy know how to find unknown side lengths nd ngle mesures in right tringles y using trigonometric functions. In this section you ll e introduced to wy of clculting the sme kind of informtion if none of the ngles of the tringle is right ngle. Such tringles re clled olique tringles. 1. Figure 9-1 shows five tringles. Ech hs sides of length 3 cm nd 4 cm. They differ in the mesure of the ngle included etween the two sides. Mesure the sides nd ngles. Do you gree with the given mesurements in ech cse?. Mesure, the third side of ech tringle. Find the length of the third side if A were 180 nd if A were 0. Record your results in tle form. 3. Store the dt from Prolem in lists on your grpher. Mke connected plot of the dt on your grpher. Given two sides nd the included ngle of tringle, find y direct mesurement the length of the third side of the tringle. 4. The plot looks like hlf-cycle of sinusoid. Find the eqution of the sinusoid tht hs the sme low nd high points nd plot it on the sme screen. Do the dt relly seem to follow sinusoidl pttern?. By the Pythgoren theorem, 3 4 if A is 90. If A is less thn 90, side is less thn, so it seems you must sutrct something from 3 4 to get the vlue of. See if you cn find wht is sutrcted! 6. Wht did you lern from doing this prolem set tht you did not know efore? PROBLEM NOTES Prolem sks students to find if A is 0. Students my hve troule understnding why the result is 1 cm rther thn 0 cm, ecuse the tringle essentilly collpses nd ppers to hve no third side. You might suggest tht students think of s the segment connecting the endpoints of the other two segments. When A is 0, the 3-cm nd 4-cm segments lie on top of one nother nd connects their right endpoints. 0 3 cm A 4 cm (cm) A A 3 cm 30 4 cm 3 cm 60 A 4 cm 3 cm A 90 4 cm 4. y 4 3 cos A (cm) A 3 cm 10 3 cm 10 A 4 cm A Figure All mesurements seem correct.. Section 9-1: Angle A cm Side 1.0 cm.1 cm 3.6 cm.0 cm 6.1 cm 6.8 cm 7.0 cm 443 No, the dt don t follow such simple sinusoid. Prolem encourges students to try to discover the lw of cosines on their own.. The formul is 3 1 4? 3? 4 cos A, tht is, 1 c c cos A. 6. Answers will vry. Section 9-1: Introduction to Olique Tringles 443

6 Section 9- PLANNING 9- Olique Tringles: The Lw of Cosines In Section 9-1, you mesured the third side of tringles for which two sides nd the included ngle were known. Think of the three tringles with included ngles 60, 90, nd 10. Clss Time 1 dy Homework Assignment RA, Q1 Q10, Prolems 1, 3, 6, 7 13 odd, 14, 1, 17, 18 Teching Resources Eplortion 9-: Derivtion of the Lw of Cosines Eplortion 9-: Angles y Lw of Cosines Supplementry Prolems Technology Resources Prolem 16: Geometric Derivtion of the Lw of Cosines Prolem Presenttion Sketch: Lw of Cosines Present.gsp Activity: Tringle nd Squres: The Lw of Cosines CAS Activity 9-: The Lw of Sines vs. the Lw of Cosines TEACHING Importnt Terms nd Concepts Lw of cosines Eplortion Notes Eplortion 9- guides students through the derivtion of the lw of cosines. Hve students work through this eplortion s you derive the lw of cosines with them. Allow out 1 minutes. See pge 446 for notes on Eplortion 9-. Ojective 444 Chpter 9: < = > For the right tringle in the middle, you cn find the third side,, using the Pythgoren theorem. 3 4 For the 60 tringle on the left, the vlue of is less thn c. For the 10 tringle on the right, is greter thn c. The eqution you ll use to find the ect length of the third side from the mesures of two sides nd the included ngle is clled the lw of cosines (ecuse it involves the cosine of the ngle). In this section you ll see why the lw of cosines is true nd how to use it. of cosines to find the length of the third side. In this eplortion you will demonstrte y mesurement tht the lw of cosines gives the correct vlue for the third side of tringle if two sides nd the included ngle re given. EXPLORATION 9-: Derivtion of the Lw of Cosines The figure shows ABC. Angle A hs een plced in stndrd position in uv-coordinte system. 1. The sides tht include ngle A hve lengths nd c. Write the coordintes of points B nd v C using, c, nd functions of ngle A. C B: (u, v) (?,? ) A 1. B: (c, 0); C: ( cos A, sin A). ( cos A c) 1 ( sin A 0) cos A c cos A 1 c 1 sin A 3. cos A c cos A 1 c 1 sin A 1 c c cos A ? 4.8?.7 cos cm. Mesurements re correct. c B u C: (u, v) (?,? ). Use the distnce formul to write the squre of the length of the third side,, in terms of, c, nd functions of ngle A. 3. Simplify the eqution in Prolem y epnding the squre. Use the Pythgoren property for cosine nd sine to simplify the terms contining cos A nd sin A. continued 6. Given two sides nd the included ngle, the squre of the side opposite the given ngle equls the sum of the squres of the two given sides minus twice the product of the two given sides nd the cosine of the included ngle. Briefly: (side) 1 (side) (side)(side) (cosine of the included ngle) 7. Answers will vry. 444 Chpter 9: Tringle Trigonometry

7 EXPLORATION, continued 4. The eqution in Prolem 3 is clled the lw of cosines. Show tht you understnd wht the lw of cosines sys y using it to clculte the length of the third side of this tringle. 4.8 cm Derivtion of the Lw of Cosines In Eplortion 9-, you demonstrted tht the squre of side of tringle cn e found y sutrcting quntity from the Pythgoren epression c. Here is why this property is true. Suppose tht the lengths of two sides, nd c, of ABC re known, s is the mesure of the included ngle, A (Figure 9-, left). A 11 C.7 cm v C (u, v) r u c B A c B (c, 0) Figure 9- If you construct uv-coordinte system with ngle A in stndrd position, s on the right in Figure 9-, then vertices B nd C hve coordintes B(c, 0) nd C(u, v). By the distnce formul, (u c ) (v 0 ) By the definitions of cosine nd sine, u cos A u cos A v sin A v sin A Section Notes In this section, students derive nd pply the lw of cosines. One gol of this section is to hve students see tht the lw of cosines, which works for ll types of tringles, is n etension of the Pythgoren theorem, which works only for right tringles.. Mesure the given sides nd ngle of the tringle in Prolem 4. Do you gree with the given mesurements? Mesure the third side. Does it gree with your clculted vlue? 6. Descrie how the unknown side in the lw of cosines is relted to the given sides nd their included ngle. Strt y writing, Given two sides nd the included ngle Wht hve you lerned s result of doing this eplortion tht you did not know efore? Section 9-: 44 Note tht the lw of cosines is presented efore the lw of sines ecuse it is more relile technique. The sign of cos A indictes whether A is otuse or cute. No such informtion cn e otined y using the lw of sines. The prolems in this section involve using the lw of cosines in cses where two sides nd n included ngle re known (SAS) or where three sides re known (SSS). It cn lso e used in the miguous cse, in which two sides nd non-included ngle re known (SSA). This cse will e discussed in Section 9-. Emple 1 pplies the lw of cosines to find missing side when two sides nd n included ngle re known. In Emple, the lw of cosines is used to find missing ngle when three sides re known. In XYZ you my wnt to encourge students to write the Z with r through it so tht the Z is not confused with. The lengths given in Emple 3 do not form tringle, so the lw of cosines yields no solution. Some students my relize immeditely tht tringle cnnot e formed ecuse the lengths do not stisfy the tringle inequlity. This section provides n ecellent opportunity to review some of the tringle concepts students lerned in geometry. Here re some topics you might discuss. The fct tht the lrgest ngle in tringle is opposite the longest side nd tht the smllest ngle is opposite the shortest side The tringle inequlity Tests for determining whether n ngle is right, cute, or otuse (see Prolem 18) Conventions for nming sides nd ngles (in some tets, ngles re denoted y the Greek letters lph, ; et, ; nd gmm, ) Section 9-: Olique Tringles: The Lw of Cosines 44

8 Differentiting Instruction Pss out the list of Chpter 9 voculry, ville t for ELL students to look up nd trnslte in their ilingul dictionries. Hve ELL students find out wht the lw of cosines is clled in their own lnguge. This will help students understnding the mening of lw in the contet of mthemtics. Hve students enter the lw of cosines in their journls three times, once with on the left side, once with, nd once with c. Figuring out the other two forms will help students lern the formul. The Reding Anlysis should e done in pirs. ELL students will proly need lnguge support on Prolems Additionl Eplortion Notes Eplortion 9- hs students first find ngles y using the lw of cosines nd then verify their results y mesuring. This ctivity cn e done y groups of students in clss or ssigned for homework. Allow out 1 minutes for this eplortion. Technology Notes Prolem 16: Geometric Derivtion of the Lw of Cosines Prolem sks students to eperiment with the Dynmic Preclculus Eplortion t in order to derive the lw of cosines. Presenttion Sketch: Lw of Cosines Present.gsp, ville t is relted to the ctivity Tringles nd Squres: The Lw of Cosines mentioned net. EXAMPLE Chpter 9: SOLUTION Sustituting these vlues for u nd v nd completing the pproprite lgeric opertions gives Notes: Activity: Tringles nd Squres: The Lw of Cosines in the Instructor s Resource Book provides nother visul proof of the lw of cosines. Students uild squre with side length equl to the length of the longest leg of n otuse tringle, nd then they compre res of different figures tht result. (u c ) (v 0 ) ( cos A c ) ( sin A 0 ) c o s A c cos A c sin A ( cos A sin A) c cos A c c c cos A PROPERTY: The Lw of Cosines In tringle ABC with sides,, nd c, c c cos A Clculte the squres. Fctor from the first nd lst terms. Use the Pythgoren property. side + side (side)(side)(cosine of included ngle)= (third side ) If the ngle mesures 90, the lw of cosines reduces to the Pythgoren theorem, ecuse cos 90 is zero. If ngle A is otuse, cos A is negtive. So you re sutrcting negtive numer from c, giving the lrger vlue for, s you found in Section 9-1. You should not jump to the conclusion tht the lw of cosines gives n esy wy to prove the Pythgoren theorem. Doing so would involve circulr resoning, ecuse the Pythgoren theorem (in the form of the distnce formul) ws used to derive the lw of cosines. A cpitl letter is used for the verte, the ngle t tht verte, or the mesure of tht ngle, whichever is pproprite. If confusion results, you cn use the symols from geometry, such s m A for the mesure of ngle A. Applictions of the Lw of Cosines You cn use the lw of cosines to clculte the mesure of either side or n ngle. In ech cse, different prts of tringle re given. Wtch for wht these givens re. In PMF, M 17, p 1.78 ft, nd f 8.4 ft. Find the mesure of the third side, m. P First, sketch the tringle nd lel the sides m nd ngles, s shown in Figure 9-. (It does f not need to e ccurte, ut it must hve the right reltionship mong sides nd ngles.) M p Figure 9- CAS Activity 9-: The Lw of Sines vs. the Lw of Cosines in the Instructor's Resource Book hs students eplore the vrious chllenges of using the lws of sines nd cosines. As students using CAS will discover, the lw of cosines is more consistent when the lgeric clcultions cn e done using grpher. Allow 0 minutes. F 446 Chpter 9: Tringle Trigonometry

9 EXAMPLE SOLUTION m (8.4)(1.78) cos 17 m m ft Use the lw of cosines for side m. In XYZ, 3 m, y 7 m, nd z 9 m. Find the mesure of the lrgest ngle. Mke sketch of the tringle nd lel the sides, Z y 7 m s shown in Figure 9-c. 3 m Y z 9 m Figure 9-c X CAS Suggestions For tringles tht cn e solved using the lw of cosines, Solve commnd is idel for differentiting etween tringles tht re impossile, possile, nd miguous. When using CAS, students cn insert vriles for ny unknown quntity nd solve the resulting eqution. The figure shows Emples 1,, nd 3. EXAMPLE 3 Too short! 3 z 11 Figure 9-d SOLUTION y 7 Recll from geometry tht the lrgest side is opposite the lrgest ngle, in this cse, Z. Use the lw of cosines with this ngle nd the two sides tht include it cos Z cos Z cos Z cos Z Z rccos( ) c o s 1 ( ) Note tht rccos( ) c o s 1 ( ) in Emple ecuse there is only one vlue of n rccosine etween 0 nd 180, the rnge of ngles possile in tringle. Suppose tht the lengths of the sides in Emple hd een 3 m, y 7 m, nd z 11 m. Wht would the mesure of ngle Z e in this cse? Write the lw of cosines for side z, the side tht is opposite ngle Z cos Z Z y cos Z Y z X cos Z 4 z = + y y cos Z 1. cos Z There is no such tringle. cos Z must e in the rnge [ 1, 1]. The geometric reson why there is no solution in Emple 3 is tht no two sides of tringle cn sum to less thn the third side. Figure 9-d illustrtes this fct. The lw of cosines signls this inconsistency lgericlly y giving cosine vlue outside the intervl [ 1, 1]. Section 9-: 447 Note tht the CAS gives oth lgeric nswers to Emple 1 even though only the positive solution mkes sense in contet. Students should lern to mke this distinction. It is helpful to restrict the ngle vlues in these emples to those suitle for tringles. This cn e done using commnd s shown in lines nd 3 of the previous figure. An lterntive use of the lw of cosines is to define Solve commnd using the generic lw of cosines formul nd sustituting for ll vlues using commnd. Some students my prefer to use this method ecuse they see the lw of cosines in its fmilir form. (In the figure the commnd is eecuted twice so tht you cn see the eginning nd end of the commnd line.) Finlly, notice tht when the side lengths re entered with units, the results lso include units. Section 9-: Olique Tringles: The Lw of Cosines 447

10 PROBLEM NOTES Supplementry prolems for this section re ville t keyonline. Remind students to store intermedite nswers without rounding. Only finl results should e rounded. Q1. i u Q. i q Q3. 1 Q4. u sin i Q. u q Q6. tn 1 q i Q7. Sinusoidl is Q8. cos 3 cos 4 1 sin 3 sin 4 Q9. Horizontl diltion y fctor of 1 Q10. sin cos Prolems 1 1 re strightforwrd nd provide prctice with the lw of cosines. In Prolems 9 nd 10, tringle cnnot e formed from the given side lengths. Students my discover this y using the tringle inequlity or y ttempting to pply the lw of cosines. 1. r 3.98 cm. d.0 in. 3. r 4.68 ft 4. k m In Prolems 1 you cn use commnd to prevent unnecessry solutions.. U G T E This is not possile tringle, ecuse This is not possile tringle, ecuse O Q 90 Prolem Set 9- Reding Anlysis From wht you hve red in this section, wht do you consider to e the min ide? How is the lw of cosines relted to the Pythgoren theorem? Wht three prts of tringle should you know in order to use the lw of cosines in its frontwrd form, nd wht prt cn you clculte using the lw? How cn you use the lw of cosines to clculte the mesure of n ngle? Quick Review Prolems Q1 Q6 refer to right tringle QUI (Figure 9-e). Q min i u U Figure 9-e 448 Chpter 9: q I Q1. cos Q? Q. tn I? Q3. sin U? Q4. In terms of side u nd ngle I, wht does i equl? Q. In terms of sides u nd q, wht does i equl? Q6. In terms of the inverse tngent function, Q?. Q7. The grph of y cos sin 1 is periodic with vrying?. Q8. In terms of cosines nd sines of 3 nd 4, cos(3 4 )?. Q9. Wht trnsformtion of y cos is epressed y y cos? Q10. Epress sin in terms of sin nd cos. For Prolems 1 4, find the length of the specified side. 1. Side r in RPM, if p 4 cm, m cm, nd R 1 Prolems 13 nd 14 re idel to do using The Geometer s Sketchpd. If the progrm is unville, students cn use protrctor, compss, nd ruler to construct the tringles nd find the unknown mesures. Students mesurements should gree with the result given y the lw of cosines. Centimeter grph pper from the lckline mster in the Instructor s Resource Book my e used.. Side d in CDE, if c 7 in., e 9 in., nd D Side r in PQR, if p 3 ft, q ft, nd R Side k in HJK, if h 8 m, j 6 m, nd K 17 For Prolems 1, find the mesure of the specified ngle.. Angle U in UMP, if u in., m 3 in., nd p 4 in. 6. Angle G in MEG, if m cm, e 6 cm, nd g 8 cm 7. Angle T in BAT, if 6 km, 7 km, nd t 1 km 8. Angle E in PEG, if p 1 ft, e ft, nd g 16 ft 9. Angle Y in GYP, if g 7 yd, y yd, nd p 13 yd 10. Angle N in GON, if g 6 mm, o 3 mm, nd n 1 mm 11. Angle O in NOD, if n 147 yd, o 03 yd, nd d 148 yd 1. Angle Q in SQR, if s 104 cm, q 46 cm, nd r 193 cm 13. Accurte Drwing Project:. Using computer softwre such s The Geometer s Sketchpd, or using ruler nd protrctor, construct RPM from Prolem 1. Then mesure side r. Does the mesured vlue gree with the clculted vlue in Prolem 1 within 0.1 cm?. Using Sketchpd or ruler, compss, nd protrctor, construct MEG from Prolem 6. Construct the longest side, 8 cm, first. Then drw n rc or circle of rdius cm from one endpoint nd n rc of rdius 6 cm from the other endpoint. The third verte is the point where the rcs intersect. Mesure ngle G. Does the mesured vlue gree with the clculted vlue in Prolem 6 within 1? 13. r 4.0 cm, p 4.0 cm, m.0 cm, R m.0 cm, e 6.0 cm, g 8.0 cm, G ft 14. $7.7 14c. $ c o s Chpter 9: Tringle Trigonometry

11 14. Fence Prolem: Mttie works for fence compny. She hs the jo of pricing fence to go cross tringulr lot t the corner of Almo nd Heights Streets, s shown in Figure 9-f. The streets intersect t 6 ngle. The lot etends 00 ft from the intersection long Almo nd 10 ft from the intersection long Heights. Figure 9-f. How long will the fence e?. How much will it cost her compny to uild the fence if fencing costs $3.7 per foot? c. Wht price should she quote to the customer if the compny is to mke 3% profit? 1. Flight Pth Prolem: Sm flies helicopter to drop supplies to strnded flood victims. He will fly from the supply depot, S, to the drop point, P. Then he will return to the helicopter s se t B, s shown in Figure 9-g. The drop point is 1 mi from the supply depot. The se is 1 mi from the drop point. It is 33 mi etween the supply depot nd the se. Becuse the return flight to the se will e mde fter drk, Sm wnts to know in wht direction to fly. Wht is the ngle etween the two pths t the drop point? S 1 mi? P 33 mi Figure 9-g 1 mi Prolem 16 gives students n opportunity to eplore geometricl derivtion of the lw of cosines using Dynmic Preclculus Eplortion t preclc. 16. Answers will vry. B 16. Geometricl Derivtion of the Lw of Cosines Prolem: Open the Lw of Cosines eplortion t Eplin in writing how this sketch provides visul verifiction of the lw of cosines. 17. Derivtion of the Lw of Cosines Prolem: Figure 9-h shows XYZ with ngle Z in stndrd position. The sides tht include ngle Z re 4 units nd units long, s shown. Find the coordintes of points X nd Y in terms of 4,, nd ngle Z. Then use the distnce formul, pproprite lger, nd trigonometry to show tht Z v z 4 4 cos Z 4 X Figure 9-h z Y 18. Acute, Right, or Otuse Prolem: The lw of cosines sttes tht in XYZ y z yz cos X. Eplin how the lw of cosines llows you to mke quick test to see whether ngle X is cute, right, or otuse, s shown in this o: u PROPERTY: Test for the Size of n Angle in Tringle In XYZ: If y z, then ngle X is n cute ngle. If y z, then ngle X is right ngle. If y z, then ngle X is n otuse ngle.. Without using your clcultor, find whether ngle X is cute, right, or otuse if 7 cm, y cm, nd z 4 cm. Section 9-: X (4 cos Z, 4 sin Z), Y (, 0), so z (4 cos Z ) 1 (4 sin Z 0 ) 4 cos Z? 4? cos Z sin Z 4 ( sin Z 1 cos Z) 1? 4? cos Z 4 1? 4? cos Z Prolem 18 poses test for determining whether n ngle of tringle is cute, right, or otuse. 18. If y 1 z, then y 1 z yz cos X y 1 z, which hppens ectly when cos X 0; so X is cute. If y 1 z, then y 1 z yz cos X y 1 z, which hppens ectly when cos X 0; so X is right. If y 1 z, then y 1 z yz cos X y 1 z, which hppens ectly when cos X 0; so X is otuse ; so X is otuse. Additionl CAS Prolems 1. In the tringles shown elow, determine the mesures of the unknown sides cm. cm 11. cm 9.3 y 10. cm 40. The perimeter of tringle is 0 units nd the length of one side is 9. units. If the ngle opposite the given side mesures rdins, wht re the lengths of the other two sides? See pge 1018 for nswers to CAS Prolems 1 nd. Section 9-: Olique Tringles: The Lw of Cosines 449

12 Section 9-3 Clss Time 1 dy PLANNING Homework Assignment RA, Q1 Q10, Prolems 1, 3, 7 9, 11, 13, 14 Teching Resources Eplortion 9-3: Are of Tringle nd Hero's Formul Eplortion 9-3: Derivtion of Hero's Formul Supplementry Prolems Technology Resources Prolem 11: Vrile Tringle Prolem Eplortion 9-3: Are of Tringle nd Hero s Formul TEACHING Importnt Terms nd Concepts Hero s formul Semiperimeter Eplortion Notes Eplortion 9-3 guides students through the derivtion of the formul Are 1 c sin A nd poses severl prolems in which students need to pply the formul. You might ssign this ctivity t the eginning of clss so tht students cn derive the formul on their own. Allow 1 0 minutes to complete this ctivity. 1. Answers should gree.. h 7 sin By mesurement, h 4.3 cm, which grees. 3. Are cm 4. Are m A Chpter 9: 9-3 Ojective Are of Tringle Recll from erlier mth clsses tht the re of tringle equls hlf the product of the se nd the ltitude. In this section you ll lern how to find this re from two side lengths nd the included ngle mesure. This is the sme informtion you use in the lw of cosines to clculte the length of the third side. Given the mesures of two sides nd the included ngle, or the mesures of ll three sides, find the re of the tringle. In this eplortion you will discover quick method for clculting the re of tringle from the mesures of two sides nd the included ngle. EXPLORATION 9-3: Are of Tringle nd Hero s Formul X For Prolems 1 3, XYZ hs sides y 8 cm nd z 7 cm nd included ngle X with mesure z 7 cm y 8 cm 1. Do you gree with the given mesurement for y? for z? for X?. Use the given mesurements to clculte ltitude h. Mesure h. Does it gree with the clcultion? 3. Recll from geometry tht the re of tringle is 1 (se)(ltitude). Find the re of XYZ. 4. By sustituting z sin X for the ltitude in Prolem 3 you get Are 1 yz sin X or, in generl, Are 1 (side)(side)(sine of included ngle) Sketch tringle with sides 43 m nd 1 m nd included ngle 143. Use this re formul to find the re of this tringle. 6. Are s ; The nswer is the sme s the re clculted in Prolem Are ft 10. Answers will vry. Y h Z For Prolems 8, ABC hs sides 8, 7, nd c 11. C 7 8 A c 11 B. Find the mesure of ngle A using the lw of cosines. Store the nswer without rounding. 6. Use the unrounded vlue of A nd the re formul of Prolem 4 to find the re of ABC. 7. Clculte the semiperimeter (hlf the perimeter) of the tringle, s 1 ( c). 8. Evlute the quntity s(s )(s )(s c). Wht do you notice out the nswer? 9. Use Hero s formul, nmely, Are s(s )(s )(s c) to find the re of this tringle. 10. Wht did you lern s result of doing this eplortion tht you did not know efore? Section Notes In this section, the trditionl formul for the re of tringle, A 1 h, is trnsformed into one involving trigonometry, A 1 c sin A. The trnsformtion is strightforwrd. The derivtion of the re formul given is for n cute ngle A. The formul lso works if A is otuse. You my wnt to show students the derivtion. 40 Chpter 9: Tringle Trigonometry

13 B c h A C Figure 9-3 EXAMPLE 1 SOLUTION EXAMPLE SOLUTION H j cm d 7 cm D h 11 cm J Figure 9-3c The following is derivtion of the re formul you discovered in Eplortion 9-3. Figure 9-3 shows ABC with se nd ltitude h. Are 1 h From geometry, re equls hlf se times ltitude. Are 1 (c sin A) h Becuse sin A c. Are 1 c sin A PROPERTY: Are of Tringle In ABC, Are 1 c sin A Verlly: The re of tringle equls hlf the product of two of its sides nd the sine of the included ngle. In ABC, 13 in., 1 in., nd C 71. Find the re of the tringle. Sketch the tringle to e sure you re given two sides nd the included ngle (Figure 9-3). Are 1 (13)(1)sin in. Find the re of JDH if j cm, d 7 cm, nd h 11 cm. Sketch the tringle to give yourself picture of wht hs to e done (Figure 9-3c). h j d jd cos H cos H j d h jd ()(7) H rccos( ) c o s 1 ( ) Store without rounding. Are 1 ()(7)sin cm Hero s Formul Use the lw of cosines to clculte n ngle mesure. Solve for cos H. It is possile to find the re of tringle directly from the lengths of three sides without going through the ngle clcultions of Emple. The method uses Hero s formul, nmed fter Hero of Alendri, who lived round 100.c.e. A B 71 1 in. Figure in. C Hero s formul is specil cse of Brhmgupt s formul, which sttes tht the re of qudrilterl inscried in circle is A (s )(s )(s c)(s d) where s is the semiperimeter nd,, c, nd d re the lengths of the four sides. Letting one side equl zero mkes the qudrilterl degenerte to tringle, nd Hero s formul ppers. (The formul is sometimes referred to s Heron s formul, lthough this is grmmticl error. Heron is the genitive (possessive) cse of Hero, so Heron lredy mens of Hero. ) Differentiting Instruction Hve students reserch the nme for Hero s formul in their own lnguge. Depending on the lnguge skills of your clss, students my e le to do Eplortion 9-3 individully or my need to do it in pirs. Hve students write the formul for the re of tringle s they hve previously lerned it in their journls, nd then hve them write ll three forms of the formul on pge 41. Finlly, hve them write the re of tringle property in their own words. Most students will e le to do the Reding Anlysis individully, ut provide ssistnce if needed. You my need to provide support with the lnguge in Prolems Hero of Alendri h B c A Are 1 h 1 [c sin(180 A)] 1 c(sin 180 cos A cos 180 sin A) Composite rgument property for sin(a B). 1 c sin A sin 180 0; cos C Section 9-3: 41 This derivtion is n idel opportunity to remind students tht n ltitude of tringle my fll outside the tringle Emple 1 involves direct sustitution into the new re formul, wheres Emple requires first finding n ngle using the lw of cosines. In Emple 3, Hero s formul, nmed fter Hero of Alendri, who lived out 100 B.C.E., is used to find the re of tringle given the length of three sides. Additionl Eplortion Notes Eplortion 9-3 requires students to pply the lw of cosines, the formul Are 1 c sin A, nd Hero s formul nd then guides them through the derivtion of Hero s formul. You might use this ctivity s tke-home ssignment or s group quiz. Allow t lest 0 minutes. Section 9-3: Are of Tringle 41

14 Technology Notes Prolem 11 sks students to find the re of tringle s function of one ngle, when the two legs creting tht ngle re of fied lengths. Students re sked to view Dynmic Preclculus Eplortion of this tringle t This prolem is relted to Eplortion 9-1 in the Instructor s Resource Book, in the sense tht oth ctivities egin with tringle tht hs two fied legs nd vrile included ngle u, nd they epress some prmeter s function of u. Eplortion 9-3 guides students through deriving formul for re, given the lengths of two sides nd the mesure of the included ngle. Sketchpd cn e used to test conjectures out formuls ginst the re s clculted y the softwre. CAS Suggestions Are prolems cn e solved similrly to the lw of cosines prolems s descried in the CAS Suggestions on pge 447 in Section 9-. One pproch to the trigonometric formul for the re of tringle is to define functions using CAS. Students cn define the re function s shown in the figure, then use the function to evlute the re. EXAMPLE 3 4 Chpter 9: SOLUTION Prolem Set 9-3 Reding Anlysis PROPERTY: Hero s Formul From wht you hve red in this section, wht do you consider to e the min ide? In wht wy is the re formul Are 1 c sin A relted to the formul Are 1 (se)(height)? Wht formul llows you to clculte the re of tringle directly, from three given side lengths? min Quick Review Prolems Q1 Q refer to Figure 9-3d. s r Q1. Stte the lw of cosines using ngle R. T Figure 9-3d Q. Stte the lw of cosines using ngle S. Q3. Stte the lw of cosines using ngle T. Q4. Epress cos T in terms of sides r, s, nd t. Q. Why do you need only the function cos 1, not the reltion rccos, when using the lw of cosines to find n ngle? Q6. When you multiply two sinusoids with very different periods, you get function with vrying?. Q7. Wht is the first step in proving tht trigonometric eqution is n identity? Q8. Which trigonometric functions re even functions? Q9. If ngle is in stndrd position, then horizontl coordinte is the definition of?. rdius If your students re using units in prolem, ut get English units when they re epecting metric, check the system settings or convert the units directly using the conversion commnd. Students my lso find it helpful to define Hero s formul s function. Use the formul for the semiperimeter insted of fourth vrile to eliminte etr clcultions. It is prticulrly nice tht the CAS will return n ect vlue. In ABC, the re is given y Are s(s )(s )(s c) where s is the semiperimeter (hlf the perimeter), 1 ( c). Find the re of JDH in Emple using Hero s formul. Confirm tht you get the sme nswer s in Emple. s 1 ( 7 11) 11. Are 11.(11. )(11. 7)( ) cm Agrees with Emple. R t S Q10. In the composite rgument properties, cos ( y)?. For Prolems 1 4, find the re of the indicted tringle. 1. ABC, if ft, 9 ft, nd C 14. ABC, if 8 m, c 4 m, nd A RST, if r 4.8 cm, t 3.7 cm, nd S XYZ, if yd, z 8.6 yd, nd Y 138 For Prolems 7, use Hero s formul to clculte the re of the tringle.. ABC, if 6 cm, 9 cm, nd c 11 cm 6. XYZ, if 0 yd, y 90 yd, nd z 100 yd 7. DEF, if d 3.7 in., e.4 in., nd f 4.1 in. 8. Comprison of Methods Prolem: Reconsider Prolems 1 nd 7.. For ABC in Prolem 1, clculte the length of the third side using the lw of cosines. Store the nswer without rounding. Then find the re using Hero s formul. Do you get the sme nswer s in Prolem 1?. For DEF in Prolem 7, clculte the mesure of ngle D using the lw of cosines. Store the nswer without rounding. Then find the re using the re formul s in Emple. Do you get the sme nswer s in Prolem 7? 4 Chpter 9: Tringle Trigonometry

15 9. Hero s Formul nd Impossile Tringles Prolem: Suppose someone tells you tht ABC hs sides cm, 6 cm, nd c 13 cm.. Eplin why there is no such tringle.. Apply Hero s formul to the given informtion. How does Hero s formul llow you to detect tht there is no such tringle? 10. Lot Are Prolem: Sen works for rel estte compny. The compny hs contrct to sell the tringulr lot t the corner of Almo nd Heights Streets (Figure 9-3e). The streets intersect t 6 ngle. The lot etends 00 ft from the intersection long Almo nd 10 ft from the intersection long Heights.. Find the re of the lot.. Lnd in this neighorhood is vlued t $3,000 per cre. An cre is 43,60 ft. How much is the lot worth? c. The rel estte compny will ern commission of 6% of the sles price. If the lot sells for wht it is worth, how much will the commission e? Figure 9-3e 11. Vrile Tringle Prolem: Figure 9-3f shows ngle in stndrd position in uv-coordinte system. The fied side of the ngle is 3 units long, nd the rotting side is 4 units long. As increses, the re of the tringle shown in the figure is function of. v 4 3 Figure 9-3f u The nmes given to the trigre nd hero functions re entirely ritrry. Some CAS grphers require the use of stndrd function nmes. Students cn nme functions in whtever wy helps them rememer est. Q1. r s 1 t st cos R Q. s r 1 t rt cos S Q3. t r 1 s rs cos T Q4. cos T r 1 s t rs. Write the re s function of.. Mke tle of vlues of re for ech 1 from 0 through 180. c. Is this sttement true or flse? The re is n incresing function of for ll ngles from 0 through 180. Give evidence to support your nswer. d. Find the domin of for which this sttement is true: The re is sinusoidl function of. Eplin why the sttement is flse outside this domin. 1. Unknown Angle Prolem: Suppose you need to construct tringle with one side 14 cm, nother side 11 cm, nd given re.. Wht two possile vlues of the included ngle will produce n re 0 cm?. Show tht there is only one possile vlue of the included ngle if the re is 77 cm. c. Show lgericlly tht there would e no possile vlue of the ngle if the re were 100 cm. 13. Compring Formuls Prolem: Demonstrte numericlly tht the re formul nd Hero s formul oth yield the sme results for tringle with hypotenuse cm. Cn you show it without using decimls? 14. Derivtion of the Are Formul Prolem: Figure 9-3g shows XYZ with ngle Z in stndrd position. The two sides tht include ngle Z re 4 nd units long. Find the ltitude h in terms of length 4 nd ngle Z. Then show tht the re of the tringle is given y Are 1 ()(4) sin Z Z v 4 X h Figure 9-3g z Y u Section 9-3: 43 Q. The other vlues re either negtive or greter thn 180 nd therefore could not e ngles of tringle. Q6. Amplitude Q7. Strt with the more complicted side nd try to simplify it to equl the other side. Q8. Cosine nd secnt Q9. cos u Q10. cos cos y sin sin y PROBLEM NOTES Supplementry prolems for this section re ville t keyonline. A CAS cn e used in Prolems 1 7 to enter units nd otin ect results insted of numeric pproimtions. Using the defined functions mentioned in the CAS Suggestions is prticulrly useful. Prolem 8 sks students to compre res clculted using Hero s formul with res clculted using Are 1 c sin A. In Prolem 8, students cn use CAS to get ect results insted of pproimtions. Students should use Boolen logic to determine whether their results re equivlent to the epected results. Prolem 11 cn e eplored dynmiclly using the Dynmic Preclculus Eplortion Vrile Tringle. In Prolem 11, students who define the trigonometric re formul s function cn grph it without resorting to tle of vlues. Define the ngle s to disply the grph. Additionl CAS Prolems 1. One side of tringle hs length nd n djcent ngle mesures. Find the lengths of the other two sides of the tringle if the tringle s re is 1.. An isosceles tringle is defined so tht the mesure of the ngle etween the congruent sides, in degrees, is numericlly equivlent to the length of the congruent sides. Wht is the mimum re of such tringle nd how long re the congruent sides? See pge for nswers to Prolems 1 14 nd CAS Prolems 1 nd. Section 9-3: Are of Tringle 43

16 Section 9-4 Clss Time 1 dy PLANNING Homework Assignment RA, Q1 Q10, Prolems 1 9 odd, 10, 11, 13, Ojective Olique Tringles: The Lw of Sines Becuse the lw of cosines involves ll three sides of tringle, you must know t lest two of the sides to use it. In this section you ll lern the lw of sines, which lets you clculte side length of tringle if only one side nd two ngles re given. Given the mesure of n ngle, the length of the side opposite this ngle, nd one other piece of informtion out tringle, find the other side lengths nd ngle mesures. In this eplortion you will use the rtio of side length to the sine of the opposite ngle to find the mesures of other prts of tringle. Teching Resources Eplortion 9-4: The Lw of Sines Eplortion 9-4: The Lw of Sines for Angles Test 4, Sections 9-1 to 9-4, Forms A nd B Technology Resources Prolem 13: Geometric Derivtion of the Lw of Sines Prolem Presenttion Sketch: Lw of Sines Present.gsp Eplortion 9-4: The Lw of Sines for Angles Activity: The Lw of Sines TEACHING Importnt Terms nd Concepts Lw of sines Eplortion Notes Eplortion 9-4 requires students to mesure sides nd ngles of tringle, verify the lw of sines numericlly, nd then use the lw of sines to find the remining prts of the tringle. After students hve worked with the lw of sines, they derive the rule lgericlly. Allow 1 0 minutes for this eplortion. See pge 4 for notes on dditionl eplortions. 1. Mesurements re correct sin ; 7.0 sin Yes, to within mesurement error cm 6. Yes 44 Chpter 9: Tringle Trigonometry EXPLORATION 9-4: The Lw of Sines Y 44 Chpter 9: c sin A; c sin B; c sin A c sin B c sin A 9. c sin B sin C c c c sin A sin B sin C c z X 1. In XYZ, re the following mesurements correct? y 6.0 cm z 7.0 cm Y 7 Z 78. Assuming tht the mesurements in Prolem 1 re correct, clculte these rtios: y sin Y z sin Z 3. The lw of sines sttes tht within tringle, the rtio of the length of side to the sine of the opposite ngle is constnt. Do the clcultions in Prolem seem to confirm this property? 4. Mesure ngle X. sin C y sin C 10. The sttements re equivlent ecuse if the prts of n eqution re equl nd nonzero, then the reciprocls of the prts of the eqution re equl to ech other nd nonzero. 11. Answers will vry. Z. Assuming tht the lw of sines is correct, sin X y sin Y Use this informtion nd the mesured vlue of X to clculte length. 6. Mesure side. Does your mesurement gree with the clculted vlue in Prolem? The lw of sines cn e derived lgericlly. A C c B 7. For ABC, use the re formul to write the re three wys:. Involving ngle A. Involving ngle B c. Involving ngle C 8. The re of tringle is independent of the wy you clculte it, so ll three epressions in Prolem 7 re equl to ech other. Write three-prt eqution epressing this fct. 9. Divide ll three sides of the eqution in Prolem 8 y whtever is necessry to leve only the sines of the ngles in the numertors. Simplify. Section Notes The lw of sines cn e pplied when two ngles nd non-included side of tringle re known (AAS) or when two ngles nd n included side re known (ASA). It cn lso e used with cution in the miguous cse, in which two sides nd non -included ngle re known (SSA). The miguous cse is covered in Section 9-. continued

17 EXPLORATION, continued A 10. The eqution you should hve gotten in Prolem 9 is the lw of sines. Eplin why it is equivlent to the lw of sines s written in Prolem. C c Figure 9-4 B In Eplortion 9-4, you demonstrted tht the lw of sines is correct nd used it to find n unknown side length of tringle from informtion out other sides nd ngles. The lw of sines cn e proved with the help of the re formul from Section 9-3. Figure 9-4 shows ABC. In the previous section you found tht the re is equl to 1 c sin A. The re is constnt no mtter which pir of sides nd included ngle you use c sin A c sin B sin C Set the res equl. c sin A c sin B sin C Multiply y. c sin A c sin B c sin C c c sin A sin B sin C c Divide y c. This finl reltionship is clled the lw of sines. If three nonzero numers re equl, then their reciprocls re equl. So you cn write the lw of sines in nother lgeric form: sin A sin B c sin C PROPERTY: The Lw of Sines In ABC, sin A sin B sin C c 11. Wht did you lern s result of doing this eplortion tht you did not know efore? nd sin A sin B c sin C Verlly: Within ny given tringle, the rtio of the sine of n ngle to the length of the side opposite tht ngle is constnt. Becuse of the different comintions of sides nd ngles for ny given tringle, it is convenient to revive some terminology from geometry. The initils SAS stnd for side, ngle, side. This mens tht s you go round the perimeter of the tringle, you re given the length of side, the mesure of n ngle, nd the length of side, in tht order. SAS is equivlent to knowing two sides nd the included ngle, the sme informtion used in the lw of cosines nd in the re formul. Similr menings re ttched to ASA, AAS, SSA, nd SSS. Although the lw of sines is n esy nd sfe wy to find side lengths, students must e creful when using it to find ngle mesures. Prolem 11 illustrtes the risks involved in using the lw of sines to find ngle mesures resulting from the fct tht there re two ngles in the intervl [0, 180 ] with given sine vlue. Be sure to discuss this prolem in clss. Eplortion 9-4 presents nother wy to pproch this prolem. 4 The inverse sine function lwys gives the first-qudrnt ngle. To find the correct nswer, students must consider the generl solution for rcsine. You might present the tles on pges to summrize wht students hve lerned so fr out finding unknown mesures in tringles. A lckline mster is ville in the Instructor's Resource Book. Section 9-4: Differentiting Instruction The tles on pges should e duplicted nd pssed out to students. Eplortion 9-4 should e done in pirs ecuse the lnguge is complicted. Eplin the mening of ssuming in Eplortion Prolems nd ; check for understnding. Students should write the lw of sines in their journls in oth forms. Prolems 9 11 will require lnguge support for ELL students. Additionl Eplortion Notes Eplortion 9-4 is worksheet for Prolem 11. If you don t wnt your students to do the prolem for homework, this eplortion provides gret opportunity for collortive effort. By doing this ctivity, students will discover the pitflls of using the lw of sines to find ngle mesures. Allow out 0 minutes for this eplortion. Technology Notes Prolem 13: The Geometric Derivtion of the Lw of Sines Prolem sks students to use Dynmic Preclculus Eplortion t nd descrie how the sketch provides proof of the lw of sines. Presenttion Sketch: Lw of Sines Present.gsp, ville t demonstrtes proof of the lw of sines. It is relted to the Lw of Sines ctivity. Section 9-4: Olique Tringles: The Lw of Sines 4

18 Technology Notes (continued) Eplortion 9-4 in the Instructor s Resource Book demonstrtes the dnger of trying to find the mesure of n ngle using the lw of sines. Sketchpd cn provide useful eplortion tool. In prticulr, one pge in the presenttion sketch Lw of Sines Present.gsp, mentioned erlier, visully demonstrtes this dnger. Activity: The Lw of Sines in the Instructor s Resource Book hs students merge two tringles to see how the sine rtios re equl. On the second pge of the provided sketch, students re sked to solve n pplied prolem using the lw of sines. Allow 40 0 minutes. CAS Suggestions Becuse the difficulty of the lger is significntly minimized, it is esy to solve mny tringle prolems with multiple unknowns using systems of equtions. For Emples 1 nd, two unknowns re sought, so two equtions re required. Using oth the lw of cosines nd the lw of sines gives students two equtions to solve s system. Although CAS will return two nswers, one of them is n impossile negtive side length. c B EXAMPLE 1 A SOLUTION Cse: AAS Figure 9-4 C EXAMPLE SOLUTION Given AAS, Find the Other Sides Emple 1 shows you how to clculte two side lengths given the third side nd two ngles. In ABC, B 64, C 38, nd 9 ft. Find the lengths of sides nd c. First, drw picture, s in Figure 9-4. Becuse you know the ngle opposite side c ut not the ngle opposite side, it s esier to strt with finding the length of side c. c sin 38 9 sin 64 c 9 sin ft sin 64 Use the lw of sines. Put the unknown in the numertor on the left side. Multiply oth sides y sin 38 to isolte c on the left. To find y the lw of sines, you need the mesure of A, the opposite ngle. A 180 (38 64 ) 78 sin 78 9 sin 64 9 sin ft sin ft nd c 6.16 ft Given ASA, Find the Other Sides The sum of the interior ngles in tringle is 180. Use the pproprite prts of the lw of sines with in the numertor. Emple shows you how to clculte side lengths if the given side is included etween the two given ngles. In ABC, 8 m, B 64, nd C 38. Find the lengths of sides nd c. First, drw picture (Figure 9-4c). The picture revels tht in this cse you do not know the ngle opposite the given side. So you clculte this ngle mesure fi r s t. From there on, it is fmilir prolem, similr to Emple 1. A c? B? m Cse: ASA Figure 9-4c A 180 (38 64 ) 78 sin 64 8 sin 78 8 sin m sin 78 C Use the pproprite prts of the lw of sines. Students cn use similr pproch to solving Prolem 11. Note tht in this cse CAS will return only one solution, nd second system is required. Students will need to oserve the wrning on pge 47 nd eliminte the A solution 46 Chpter 9: Wht You Are Given Wht You Wnt to Find Lw to Apply Three sides (SSS) An unknown ngle Lw of cosines Two sides nd n included The unknown side Lw of cosines ngle (SAS) Two ngles nd n included side (ASA) An unknown side Lw of sines (must first find missing ngle, using 180 (A 1 B)) 46 Chpter 9: Tringle Trigonometry

19 c sin 38 8 sin 78 c 8 sin m sin m nd c.04 m Use the pproprite prts of the lw of sines. ecuse no tringle cn contin two otuse ngles. Prolem Set 9-4 Reding Anlysis The Lw of Sines for Angles From wht you hve red in this section, wht do you consider to e the min ide? Bsed on the verl sttement of the lw of sines, why is it necessry to know t lest one ngle in the tringle to use the lw? In the solution to Emple 1, why is it dvisle to put the unknown side length in the numertor on the left side of the eqution? Why cn you e led to n incorrect nswer if you try to use the lw of sines to find n ngle mesure? Quick Review Q1. Stte the lw of cosines for PAF involving ngle P. Q. Stte the formul for the re of PAF involving ngle P. Q3. Write two vlues of rcsin 0. tht lie etween 0 nd 180. Q4. If sin , then sin( )?. Q. cos 6 A. 1 B. 1 C. 3 3 D. 3 E. 3 min You cn use the lw of sines to find n unknown ngle of tringle. However, you must e creful ecuse there re two vlues of the inverse sine reltion etween 0 nd 180, either of which could e the nswer. For instnce, rcsin or ; oth could e ngles of tringle. Prolem 11 shows you wht to do in this sitution. Q6. A(n)? tringle hs no equl sides nd no equl ngles. Q7. A(n)? tringle hs no right ngle. Q8. Stte the Pythgoren property for cosine nd sine. Q9. cos cos( )? in terms of cosines nd sines of. Q10. The mplitude of the sinusoid y 3 4 cos ( 6) is?. 1. In ABC, A, B 31, nd 8 cm. Find the lengths of side nd side c.. In PQR, P 13, Q 133, nd q 9 in. Find the lengths of side p nd side r. 3. In AHS, A 7, H 109, nd 10 yd. Find the lengths of side h nd side s. 4. In BIG, B, I 79, nd 0 km. Find the lengths of side i nd side g.. In PAF, P 8, f 6 m, nd A 117. Find the lengths of side nd side p. 6. In JAW, J 48, ft, nd W 73. Find the lengths of side j nd side w. 7. In ALP, A 8, p 30 ft, nd L 87. Find the lengths of side nd side l. 8. In LOW, L, o 00 m, nd W 3. Find the lengths of side l nd side w. Section 9-4: Wht You Are Given Wht You Wnt to Find Lw to Apply Two ngles nd An unknown side Lw of sines non-included side (AAS) Two ngles nd non-included side (AAS) The unknown ngle 180 (A 1 B) 47 When solving systems of equtions involving trigonometric functions, it is criticl for students to restrict vrile ngles to their known vlues. Notice tht this is done t the ends of oth lines in the previous imge. PROBLEM NOTES Q1. p 1 f f cos P Q. 1 f sin P Q3. 30, 10 Q Q. D Q6. Sclene Q7. Olique Q8. sin u 1 cos u 1 Q9. cos sin Q10. 4 Prolems 1 8 re firly strightforwrd nd re similr to Emples 1 nd. Students cn use system of equtions on CAS s descried in the CAS Suggestions to solve Prolems 1 8. This method enles students to otin the vlues of oth side lengths using single commnd cm ; c cm. p.77 in.; r 6.88 in. 3. h 49.9 yd; s yd 4. i 6. km; g 66.0 km. 9.3 m; p 4.91 m 6. j 4.33 ft; w.8 ft ft; l 1.6 ft 8. l 00.1 m; w m Section 9-4: Olique Tringles: The Lw of Sines 47

20 Prolem Notes (continued) In Prolem 9, students will need to recognize tht the side of length 1000 is opposite the lrgest ngle, nd the other two sides ech must e less thn m; y m 9. $67,0.70 9c. $10,41.0 over y nd $38,00.3 over ft ft 10c. It is fster to retrce the originl route. Prolem 11 is well worth spending time on. It helps students understnd the pitflls of using the lw of sines to find ngle mesures. Eplortion 9-4 covers the sme content. 11. A C c. C d. This is the complement of nd one of the generl vlues of rcsin 10 sin A. 7 11e. The principl vlues of rccos go from 0 to 180 ; negtive rgument will give n otuse ngle nd positive rgument will give n cute ngle, lwys the ctul ngle in the tringle. But the principl vlues of rcsin go from 90 to 90 ; negtive rgument will never hppen in tringle prolem, ut positive rgument will only give n cute ngle, wheres the ctul ngle in the tringle my e the otuse complement of the cute ngle. 1. The mesured vlue should e within 0.1 of cm. 13. Answers will vry. See pge 1019 for nswers to CAS Prolems 1 nd. 9. Islnd Bridge Prolem: Suppose tht you work for construction compny tht is plnning to uild ridge from the lnd to point on n islnd in lke (Figure 9-4d). The only two plces on the lnd to strt the ridge re point X nd point Y, 1000 m prt. Point X hs etter ccess to the lke ut is frther from the islnd thn point Y. To help decide etween X nd Y, you need the precise lengths of the two possile ridges. From point X you mesure 4 ngle to the point on the islnd, nd from point Y you mesure 8 ngle. Z Islnd Bridge? X m Figure 9-4d 48 Chpter 9: Bridge? Lke Y Lnd. How long would ech ridge e?. If constructing the ridge costs $370 per meter, how much could e sved y constructing the shorter ridge? c. How much could e sved y constructing the shortest possile ridge (if tht were oky)? 10. Wlking Prolem: Amos wlks 800 ft long the sidewlk net to field. Then he turns t n ngle of 43 to the sidewlk nd heds cross the field (Figure 9-4e). When he stops, he looks ck t the strting point, finding 9 ngle etween his pth cross the field nd the direct route ck to the strting point. Figure 9-4e. How fr cross the field did Amos wlk?. How fr does he hve to wlk to go directly ck to the strting point? c. Amos wlks ft/s on the sidewlk ut only 3 ft/s cross the field. Which wy is quicker for him to return to the strting point y going directly cross the field or y retrcing the originl route? 14. A 1 y sin Z 1 yz sin X 1 z sin Y So 1 y sin Z 1 yz sin X, sin Z z sin X, sin X z, nd similrly, sin Z sin X y sin Y. 11. Lw of Sines for Angles Prolem: You cn use the lw of sines to find n unknown ngle mesure, ut the technique is risky. Suppose tht ABC hs sides 4 cm, 7 cm, nd 10 cm, s shown in Figure 9-4f. 4 cm? A C?? 7 cm 10 cm Figure 9-4f B. Use the lw of cosines to find the mesure of ngle A.. Use the nswer to prt (don t round off) nd the lw of sines to find the mesure of ngle C. c. Find the mesure of ngle C gin, using the lw of cosines nd the given side lengths. d. Your nswers to prts nd c proly do not gree. Show tht you cn get the correct nswer from your work with the lw of sines in prt y considering the generl solution for rcsine. e. Why is it dngerous to use the lw of sines to find n ngle mesure ut not dngerous to use the lw of cosines? 1. Accurte Drwing Prolem: Using computer softwre such s The Geometer s Sketchpd, or using ruler nd protrctor with pencil nd pper, construct tringle with se 10.0 cm nd se ngles 40 nd 30. Mesure the length of the side opposite the 30 ngle. Then clculte its length using the lw of sines. Your mesured vlue should e within 0.1 cm, of the clculted vlue. 13. Geometric Derivtion of the Lw of Sines Prolem: Open the Lw of Sines eplortion t Eplin in writing how this sketch provides visul verifiction of the lw of sines. 14. Algeric Derivtion of the Lw of Sines Prolem: Derive the lw of sines lgericlly. If you cnnot do it from memory, consult the tet long enough to get strted. Then try finishing on your own. Additionl CAS Prolems 1. In CBX,, c 4, nd the mesure of ngle B is 30. Determine the length of side nd the mesure of ngle C.. If XYZ hs re 10, y, nd the mesure of ngle Z is 40, find the mesure of ngle Y nd the length of side. 48 Chpter 9: Tringle Trigonometry

21 9- Ojective The Amiguous Cse From one end of long segment, you drw n 80-cm segment t 6 ngle. From the other end of the 80-cm segment, you drw 0-cm segment, completing tringle. Figure 9- shows the two possile tringles you might crete. 80 cm 0 cm 6 Long segment Figure 9-80 cm 6 0 cm Long segment Figure 9- shows why there re two possile tringles. A 0-cm rc drwn from the upper verte cuts the long segment in two plces. Ech point could e the third verte of the tringle. 80 cm 0 cm 0 cm 6 Two possile points Figure 9- Strting segment As you go round the perimeter of the tringle in Figure 9-, the given informtion is side, nother side, nd n ngle (SSA). Becuse there re two possile tringles tht hve these specifictions, SSA is clled the miguous cse. Given two sides nd non-included ngle, clculte the possile lengths of the third side. Section 9- Clss Time 1 dy PLANNING Homework Assignment RA, Q1 Q10, Prolems 1 13 odd, 14 Teching Resources Eplortion 9-: The Amiguous Cse, SSA Eplortion 9-: Golf Bll Prolem Supplementry Prolems Technology Resources Eplortion 9-: The Amiguous Cse, SSA Eplortion 9-: Golf Bll Prolem CAS Activity 9-: An Alterntive to the Lws of Sines nd Cosines TEACHING EXAMPLE 1 SOLUTION In XYZ, 0 cm, z 80 cm, nd X 6, s in Figure 9-. Find the possile lengths of side y. Sketch tringle nd lel the given sides nd ngle (Figure 9-c). Y Y X Eplortion Notes z 80 cm 6 y? Figure 9-c 0 cm Eplortion 9- llows students to investigte the SSA cse y mesuring nd drwing. You could use this ctivity s follow-up to Emple 1 to reinforce the concept of miguity nd the use of the qudrtic formul. Or you could complete it s whole-clss ctivity in plce of Emple 1. Allow 0 minutes for this eplortion. Z X y =? Z y is the unknown. I know the other two sides, ut not ngle Y! Section 9-: 49 Eplortion 9- (inspired y Chris Sollrs) shows students rel-world sitution involving the miguous cse for nlyzing the position of golf ll. Allow 0 minutes for this eplortion. Importnt Terms nd Concepts Amiguous cse Displcement Directed distnce Section Notes From their geometry courses, students my recll tht SSA is not congruence theorem. So it should not surprise them tht there re difficulties ssocited with the SSA cse. Indeed, given two sides nd non-included ngle, it my e possile to form one tringle, two tringles, or no tringle t ll. The lw-of-cosines technique demonstrted in Emple 1 is more powerful nd direct wy to find the third side length in the SSA cse thn the more trditionl technique of finding ngles first y using the lw of sines. Section 9-: The Amiguous Cse 49

22 Section Notes (continued) The computtions involved in thequdrtic formul cn e done esily on grpher. If students do not hve qudrtic formul progrm in their grphers, you might suggest tht they write or downlod one so tht they cn solve the prolems efficiently. Using the qudrtic formul is preferle to using the solver feture ecuse it is esy to miss one of the solutions with the solver feture. It is worthwhile to go through the steps of the qudrtic formul for the cses illustrted in Figures 9-d nd 9-e so tht students cn see how the solutions relte to the figures. The SSA digrms re lwys drwn prticulr wy tht mkes it esier for students to see whether or not two tringles re possiility. The given ngle is plced on the lower left with one side of the ngle on the horizontl. Move clockwise from tht verte long the given side tht is not opposite the known ngle. This tkes you to verte tht cts like pivot point for the side opposite the given ngle. Sketch this side rememering tht the side will swing out (wy from the given ngle) or possily swing in (towrd the given ngle). See Figure 9-. Encourge your students to drw their figures the sme wy. Hve volunteer drw ABC on the ord or overhed with A 30, 10, nd. Hopefully, students will see tht the tringle is right tringle, ecuse is hlf of 10 nd the tringle hs 30 ngle. Net, sk volunteer to drw ABC with A 30, 10, nd 4. (The student cn use A nd side from the previous drwing.) Students should oserve tht side is too short nd thus tht no tringle meets the conditions. Net, sk volunteer to drw ABC with A 30, 10, nd 6. There re 460 Chpter 9: Using the lw of sines to find y would require severl steps. Here is shorter method, using the lw of cosines. 0 y 8 0 y 80 cos 6 Write the lw of cosines for the known ngle, X 6. This is qudrtic eqution in the vrile y. You cn solve it using the qudrtic formul. y (160 cos 6 )y Mke one side equl zero. y ( 160 cos 6 )y Get the form y y c 0. y 160 cos 6 ( 160 cos 6 ) Use the qudrtic formul: y 4 c. y or y 107. cm or 36.3 cm You my e surprised if you use different lengths for side in Emple 1. Figures 9-d nd 9-e show this side s 90 cm nd 30 cm, respectively, insted of 0 cm. In the first cse, there is only one possile tringle. In the second cse, there is none cm 80 cm two possile tringles in this cse, ecuse side cn swing off point C towrd A or wy from A. If students don t notice tht two tringles cn e drwn, sk them if it is possile to drw second tringle meeting the conditions. Finlly, sk volunteer to drw ABC with A 30, 10, nd 11. In this cse, side cn only swing out wy from A, so there is only one tringle tht meets these conditions cm 90 cm 80 cm 30 cm 6 Strting segment Figure 9-d Figure 9-e Misses! The qudrtic formul technique of Emple 1 detects oth of these results. For 30 cm, the discriminnt, 4c, equls , mening there re no rel solutions to the eqution nd thus no tringle. For 90 cm, y or Although cnnot e side mesure of tringle, it does equl the displcement (the directed distnce) to the point where the rc would cut the strting segment if this segment were etended in the other direction. Differentiting Instruction Remind students tht two tringles cnnot e proved congruent y SSA, nd use Figure 9- to illustrte the mening of miguous cse. Go over Emple 1 nd the eplntory tet on pge 460. Check crefully for understnding. Students my need support with the lnguge in Prolems 9 nd Chpter 9: Tringle Trigonometry

23 Prolem Set 9- Reding Anlysis From wht you hve red in this section, wht do you consider to e the min ide? Sketch tringle with two given sides nd given non-included ngle tht illustrtes tht there cn e two different tringles with the sme given informtion. How cn the lw of cosines e pplied in the miguous cse to find oth possile lengths of the third side with the sme computtion? min Quick Review Prolems Q1 Q6 refer to the tringle in Figure 9-f Figure 9-f Q1. The initils SAS stnd for?. Q. Find the length of the third side. Q3. Wht method did you use in Prolem Q? Q4. Find the re of this tringle. Q. The lrgest ngle in this tringle is opposite the? side. Q6. The sum of the ngle mesures in this tringle is?. Q7. Find the mplitude of the sinusoid y 4 cos 3 sin. Q8. The period of the circulr function y 3 7 cos 8 ( 1) is A. 16 B. 8 C. D. 7 E. 3 8 Q9. The vlue of the inverse circulr function sin 1 0. is?. Q10. A vlue of the inverse circulr reltion rcsin 0. etween nd is?. For Prolems 1 8, find the possile lengths of the indicted side. 1. In ABC, B 34, 4 cm, nd 3 cm. Find c. Technology Notes Eplortion 9- in the Instructor s Resource Book sks students to investigte the question of whether knowing two side lengths nd nonincluded ngle determines tringle. Sketchpd cn e useful tool in creting constructions.. In XYZ, X 13, 1 ft, nd y ft. Find z. 3. In ABC, B 34, 4 cm, nd cm. Find c. 4. In XYZ, X 13, 1 ft, nd y 1 ft. Find z.. In ABC, B 34, 4 cm, nd cm. Find c. 6. In XYZ, X 13, 1 ft, nd y 60 ft. Find z. 7. In RST, R 130, r 0 in., nd t 16 in. Find s. 8. In OBT, O 170, o 19 m, nd t 11 m. Find. 9. Rdio Sttion Prolem: Rdio sttion KROK plns to rodcst rock music to people on the ech ner Ocen City (O.C. in Figure 9-g). Mesurements show tht Ocen City is 0 mi from KROK, t n ngle 0 north of west. KROK s rodcst rnge is 30 mi. Ocen Lnd O.C. 0 mi 0 North How fr? Angle KROK Figure 9-g 30 mi Est. Use the lw of cosines to clculte how fr long the ech to the est of Ocen City people cn her KROK.. There re two nswers to prt. Show tht oth nswers hve mening in the rel world. c. KROK plns to rodcst only in n ngle etween line from the sttion through Ocen City nd line from the sttion through the point on the ech frthest to the est of Ocen City tht people cn her the sttion. Wht is the mesure of this ngle? Section 9-: 461 Eplortion 9- in the Instructor s Resource Book demonstrtes the miguity of SSA in the contet of golf gme. Sketchpd cn e useful for constructing model. CAS Activity 9-: An Alterntive to the Lws of Sines nd Cosines in the Instructor's Resource Book presents n lternte method for finding unknown sides nd ngles in tringle. Allow 30 minutes. PROBLEM NOTES Supplementry prolems for this section re ville t keyonline. Q1. Side-Angle-Side Q. 4.7 Q3. The lw of cosines Q Q. Longest Q Q7. Q8. A Q9. p 6 Q10. p 6 Prolems 1 8 re strightforwrd prolems in which students must find the unknown side length in SSA situtions. Note tht when there is only one possile otuse tringle, the lw of cosines produces positive solution nd negtive solution. The negtive solution hs geometric mening s directed distnce. 1. c.3... cm or cm. z 16.8 ft 3. c 7.79 cm 4. z 6.13 ft or 3.10 ft. No solution. 6. No solution. 7.. in m mi. 9. The other nswer is 1.94 mi. This mens 1.94 miles to the west of Ocen City. 9c Prolems 9 13 require students to use the lw of sines to find missing ngles in the SSA cse. Students must determine eforehnd whether one or two tringles meet the given conditions. Remind students to eercise cution when using the lw of sines to find ngles. Considering the generl solution of rcsine gives two possile ngles (the sin 1 vlue nd its supplement). Students must determine whether one or oth ngles stisfy the prolem. Section 9-: The Amiguous Cse 461

24 Prolem Notes (continued) 10. C 3.00 or S Z G 7.7 Prolem 14 offers nice summry of the si different possiilities SSA prolems present. This is good prolem to discuss in clss if you do not wnt to ssign it for homework. 14. = y sin X y 14. y sin X y 14c. y sin X y 14d. y sin X y 14e. y sin X y 14f. y sin X y For Prolems 10 13, use the lw of sines to find the indicted ngle mesure. Determine eforehnd whether there re two possile ngles or just one. 10. In ABC, A 19, mi, nd c 30 mi. Find C. 11. In HSC, H 8, h 0 mm, nd c 0 mm. Find S. 1. In XYZ, X 8, 9.3 cm, nd z 7. cm. Find Z. 13. In BIG, B 110, 1000 yd, nd g 900 yd. Find G. 14. Si SSA Possiilities Prolem: Prts through f show si possiilities of XYZ if ngle X nd sides nd y re given. For ech cse, eplin the reltionship mong, y, nd the quntity y sin X. Z. X cute y No tringle X X cute One tringle y Z y sin X c. d e. f. y X cute Two tringles X cute One tringle y Z X Y 1 Y X Z Z y Z X y X otuse One tringle z Y X otuse No tringle X z Y X z Y 46 Chpter 9: Additionl CAS Prolems 1. Suppose the sides of tringle form n rithmetic sequence. If one of the ngles is right nd the lengths of ll sides re integers, wht re the lengths of the sides?. In XYZ, y 10, the mesure of ngle X is 30, nd side z is units longer thn side.. Is this ever n miguous tringle?. Under wht conditions for is there only one tringle possile? See pge 1019 for nswers to CAS Prolems 1 nd. 46 Chpter 9: Tringle Trigonometry

25 9-6 Ojective Vector Addition Suppose you strt t the corner of room nd wlk 10 ft t n ngle of 70 to one of the wlls (Figure 9-6). Then you turn 80 clockwise nd wlk nother 7 ft. If you hd wlked stright from the corner to your stopping point, how fr nd in wht direction would you hve wlked? Figure 9-6 The two motions descried re clled displcements. They re vector quntities tht hve oth mgnitude (size) nd direction (ngle). Vector quntities re represented y directed line segments clled vectors. A quntity such s distnce, time, or volume tht hs no direction is clled sclr quntity. Given two vectors, dd them to find the resultnt vector. In this eplortion you will use the properties of tringles to dd vectors. Section 9-6 Clss Time dys PLANNING Homework Assignment Dy 1: RA, Q1 Q10, Prolems 1, 3,, 7, 9 Dy : Prolems 13 1, 17, 19, 1,, 4 Teching Resources Eplortion 9-6: Sum of Two Displcement Vectors Eplortion 9-6: Nvigtion Vectors Supplementry Prolems Technology Resources Eplortion 9-6: Sum of Two Displcement Vectors EXPLORATION 9-6: Sum of Two Displcement Vectors 1. The figure shows two vectors strting from the origin. One ends t the point (4, 7), nd the other ends t the point (, 3). Copy the figure on grph pper nd trnslte one of the two vectors so tht the eginning of the trnslted vector is t the end of the other vector. Then drw the resultnt vector the sum of the two vectors. y 10 Eplortion Notes 10 Eplortion 9-6 egins y hving students use tringle properties to find the sum of two vectors. Students re then led to discover n esier wy to find the sum, using components. This is good preview of the ides in this section. Allow students 1 minutes to complete the eplortion. See pge 468 for notes on dditionl eplortions.. Clculte the length of the resultnt vector in Prolem 1 nd the ngle it mkes with the -is. 3. The two given vectors nd the resultnt vector form tringle. Clculte the mesure of the lrgest ngle in this tringle. 4. Clculte the mesure of the ngle etween the two vectors when they re plced til-totil, s they were given in Prolem 1.. In Prolem 1, you trnslted one of the vectors. Show on your copy of the figure tht you would hve gotten the sme resultnt vector if you hd trnslted the other vector. Use different color thn you used in Prolem 1. continued Section 9-6: 463. r ; u u The -component of the resultnt vector is the sum of the -components of the given vectors, nd the y -component is the sum of the y -components. 8. Answers will vry. TEACHING Importnt Terms nd Concepts Displcement vector Vector quntity Mgnitude of vector Direction of vector Vector Sclr quntity Hed of vector Til of vector Asolute vlue Equl vectors Trnslte vector Sum of two vectors Resultnt vector Unit vector Components of vector Resolving vector Bering Opposite of vector Commuttive Associtive Zero vector Closed See pge for nswers to Eplortion Prolems 1,, nd 6. Section 9-6: Vector Addition 463

26 Section Notes This section introduces students to vectors in geometricl contet tht lys the foundtion for the work with three-dimensionl vectors in Chpter 1. This section contins prolems tht re pplictions of the lw of cosines, nd it lso introduces vector components to solve the prolems in more direct mnner. Theoreticl concepts such s commuttivity nd ssocitivity re touched on in the prolem set (Prolems 6). It is recommended tht you spend two dys on this section. Cover Emples 1 nd on the first dy nd the remining emples on the second dy. This section introduces mny new terms. Encourge students to use correct terminology when discussing nd writing out vectors. Emphsize to students tht vector hs direction nd mgnitude ut does not hve specific loction. This mkes it possile to trnslte vectors in order to dd or sutrct them. (The only eception to the rule tht vector does not hve loction is the position vector which hs its til t the origin.) Emple 1 shows how to use tringle trigonometry to find the displcement tht results from comining two motions. The resultnt displcement is the vector from the eginning of the vector representing the first motion to the end of the vector representing the second motion. This resultnt vector is the sum of the two motion vectors. Hving student ct out the scenrio in this emple might help students gin insight into the resultnt vector concept. Emple 1 leds to the definition of the sum of the two vectors. Discuss the definition with students nd illustrte it with drwing. EXPLORATION, continued Beginning (til) 6. The vectors in Prolem 1 hve components in the -direction nd in the y-direction. These components re horizontl vector nd verticl vector tht cn e dded together to equl the given vector. On your grph from Prolem 1, show how the components of the longer vector cn e dded to give tht vector. A vector 464 Chpter 9: End (hed) Mgnitude (Length) (Asolute vlue) Figure 9-6 c Equl vectors Figure 9-6c EXAMPLE 1 A SOLUTION Figure 9-6d Sum of two vectors If the two vectors to e dded re not lredy hed-to-til, you cn trnslte one of them so tht they re. (The net figure shows n emple.) Th e length of directed line segment represents the mgnitude of the vector quntity, nd the direction of the segment represents the vector s direction. An rrowhed on vector distinguishes the end (its hed) from the eginning (its til), s shown in Figure 9-6. A vrile used for vector hs smll rrow over the top of it, like this,, to distinguish it A typhoon s wind speed cn rech up to 10 mi/h. from sclr. The mgnitude of the vector is lso clled its solute vlue nd is written. Vectors re equl if they hve the sme mgnitude nd the sme direction. Vectors,, c in Figure 9-6c re equl vectors, even though they strt nd end t different plces. So you cn trnslte vector without chnging its vlue. DEFINITION: Vector A vector, v, is directed line segment. The solute vlue, or mgnitude, of vector, v, is sclr quntity equl to its length. Two vectors re equl if nd only if they hve the sme mgnitude nd the sme direction. You strt t the corner of room nd wlk s shown in Figure 9-6. Find the displcement tht results from the two motions. Drw digrm showing the two given vectors nd the displcement tht results, (Figure 9-6d). They form tringle with sides 10 ft nd 7 ft nd included ngle 100 ( ) (10)(7) cos Use the lw of cosines ft 7. Give n esy wy to get the components of the resultnt vector of the two given vectors in Prolem Wht did you lern s result of doing this eplortion tht you did not know efore? Store without rounding for use lter. Trnsltion implies tht the vector is moved without chnging its mgnitude or direction. Remind students tht ny two vectors with the sme mgnitude nd direction re equl, so trnslting vector does not chnge its vlue. Emple shows how til-to-til prolem cn e trnslted into hed-to-til prolem nd then solved y tringle methods. 464 Chpter 9: Tringle Trigonometry

27 (10)( ) cos A Use the lw of cosines to find A. cos A (10)( ) A The vector representing the resultnt displcement is pproimtely 13. ft t n ngle of 38.4 to the wll. This emple leds to the grphicl definition of vector ddition. If the til of one vector is plced t the hed of nother vector, the sum of two vectors goes from the eginning of the first vector to the end of the second, representing the resultnt displcement. Becuse of this, the sum of two vectors is lso clled the resultnt vector. DEFINITION: Vector Addition Th e sum is the vector from the eginning of to the end of if the til of is plced t the hed of. Discuss the irplne sitution on pge 466. In this emple, the horizontl nd verticl velocities re written s sclr multiples of i nd j, respectively, nd then these horizontl nd verticl vectors re dded to get the resultnt velocity vector. Then eplin tht ny vector cn e written s the sum of horizontl vector nd verticl vector. For emple, if you know plne s speed nd ngle of clim, you cn clculte the clim velocity nd the ground velocity nd then write the velocity vector s the sum of horizontl nd verticl components. EXAMPLE SOLUTION Trnslte hed-to-til. 4 kt 0 Resultnt, v 9 kt A 4 kt 130 Figure 9-6e Two vectors Emple shows how to dd two vectors tht re not yet hed-to-til, using velocity vectors, for which the mgnitude is the sclr speed. A ship ner the cost is going 9 knots t n ngle of 130 to current of 4 knots. (A knot, kt, is nuticl mile per hour, slightly fster thn regulr mile per hour.) Wht is the ship s resultnt velocity with respect to the shore? Drw digrm showing two vectors 9 nd 4 units long, til-to-til, mking n ngle of 130 with ech other, s shown in Figure 9-6e. Trnslte one of the vectors so tht the two vectors re hed-to-til. Drw the resultnt vector, v, from the eginning (til) of the first to the end (hed) of the second. In its new position, the 4-kt vector is prllel to its originl position. The 9-kt vector is trnsversl cutting two prllel lines. So the ngle etween the vectors forming the tringle shown in Figure 9-6e is the supplement of the given 130 ngle, nmely, 0. From here on the prolem is like Emple 1. v kt y y A v 7.1 kt t out 104. to the current y Trnslte hed-to-til. Section 9-6: 46 Introduce the unit vectors i nd j to students, nd eplin tht ny horizontl vector cn e written s sclr multiple of i nd tht ny verticl vector cn e written s sclr multiple of j. (See the end of the Section Notes for more informtion out vector sutrction nd sclr multiples of vector.) Clim velocity Velocity Ground velocity The process of resolving vector into horizontl nd verticl components is illustrted in Emple 3. The sum of the components is equl to the originl vector. The emple leds to generl property for resolving vector into its components. Note tht some tets represent vectors s ordered pirs. The nottion v i 1 y j used in this tet is conceptully esier for most students. It reminds them tht vector is the sum of horizontl vector nd verticl vector. When vectors re written in terms of their components, you cn dd them y dding the correspond ing components. This is illustrted in Figure 9-6h. Emple 4 shows how to dd two vectors y first resolving them into components. In prt, emphsize the importnce of mking drwing to determine which vlue of rctn to choose. y Sum goes from eginning of first to end of lst. Section 9-6: Vector Addition 46

28 Section Notes (continued) You might lso work with the clss to find the sum in Emple 4 y using tringle techniques. Most students will find tht dding the vectors y first resolving them into components is the esier method. Emple investigtes nvigtion prolem. Mke sure students understnd tht ering is mesured clockwise from north rther thn counterclockwise from the positive horizontl is. When solving nvigtion prolems, some students my e more comfortle chnging the erings to stndrd ngle mesures. The solution for Emple uses tringle methods, ut you my wnt to redo it using components. (For more prctice using components to solve nvigtion prolems, see Eplortion 9-6.) Addition is the only vector opertion covered in this chpter. You my lso wnt to show students how to sutrct vectors nd how to multiply vector y sclr, topics covered in Section 1-. EXAMPLE 3 Vector Addition y Components Suppose tht n irplne is climing with horizontl velocity of 300 mi/h nd verticl velocity of 170 mi/h (Figure 9-6f). Let i nd j e unit vectors in the horizontl nd verticl directions, respectively. This mens tht ech vector hs mgnitude 1 mi/h. Verticl velocity component 170j y, verticl speed Velocity v Figure 9-6f 300i Horizontl velocity component, horizontl speed You cn write the resultnt velocity vector, v, s the sum v 300 i 170 j The 300 i nd 170 j re clled the horizontl nd verticl components of v. The product of sclr, 300, nd the unit vector i is vector in the sme direction s the unit vector ut 300 times s long. Vector hs mgnitude 3 nd direction 143 from the horizontl (Figure 9-6g). Resolve into horizontl nd verticl components. (, y) i yj Figure 9-6g Sutrcting Vectors SOLUTION e the point t the hed of Let (, y ). Using reference tringle for 143, Remind students tht you sutrct numer y dding its opposite. For emple, 1 (). In the sme wy, you sutrct vector y dding its opposite. The opposite of vector is vector with the sme mgnitude tht points in the opposite direction. The figure illustrtes difference of two vectors. 3 cos 143 nd y sin cos nd y 3 sin i 1.80 j Note tht multiplying vector y negtive numer, such s.396 i in Emple 3, gives vector tht points in the opposite direction. Vector sutrction ( ) 466 Chpter 9: DEFINITION: Vector Sutrction The opposite of, written, is vector of the sme mgnitude s tht points in the opposite direction. The difference is the sum 1 ( ). If vectors re written s the sum of components, you cn sutrct them y sutrcting correspond ing components. You cn illustrte this y revisiting Emple 4 nd finding the difference s the sum of two components. Multiplying Vector y Sclr When you dd the rel numer to itself, you get twice tht numer Chpter 9: Tringle Trigonometry

29 EXAMPLE 4 SOLUTION Emple 3 demonstrtes the following property. PROPERTY: Components of Vector If v is vector in the direction in stndrd position, then v i y j where v cos nd y v sin. Components mke it esy to dd two vectors. As shown in Figure 9-6h, if r is the resultnt vector of nd, then the components of r re the sums of the components of nd. Becuse the two horizontl components hve the sme direction, you cn dd them simply y dding their coefficients. The sme is true for the verticl components. 70 cos 70 i r sin 70 j Figure 9-6h sin j cos i Vector hs mgnitude t 70, nd hs mgnitude 6 t (Figure 9-6h). Find the resultnt vector, r, s. The sum of two components. A mgnitude nd direction ngle. r ( cos 70 ) i ( sin 70 ) j (6 cos ) i (6 sin ) j ( cos 70 6 cos ) i ( sin 70 6 sin ) j i j 7.1 i 7.3 j. r ( ) ( ) Write the components. Comine like terms. Round the fi n l nswer. By the Pythgoren theorem. rctn n Pick n 0. r t 4.34 It is resonle to sy tht when you dd vector to itself, you get twice tht vector. 1 When you multiply rel numer y 1, you get the opposite of tht numer. 1 In similr wy, when you multiply vector y 1, you get the opposite of tht vector. 1 Round the fi n l nswer. Section 9-6: 467 This resoning leds to the definition of the product of sclr (tht is, rel numer) nd vector. Sclr times vector 1.3 DEFINITION: Product of Sclr nd Vector The product is vector in the direction of if is positive nd in the direction of if is negtive. The mgnitude of the product is the mgnitude of times the solute vlue of. Differentiting Instruction Check to see whether ny of your students hve lerned different nottion for vectors thn the one used in the tet. If so, llow them to use either nottion. ELL students should work on Eplortion 9-6 in pirs; the lnguge is more complicted thn it ppers. Emple uses lnguge tht is proly unfmilir to mny students. Check crefully for understnding. Monitor students understnding of the new mteril on vector ddition y components nd components of vector, on pges The nvigtion prolems introduced on pge 468 present severl chllenges for students. The position of ering 0 is different from 0 ngle in stndrd position. Also, the voculry in nvigtion prolems is not commonly used in converstion. You might work through homework prolem, such s Prolem 17, s nother emple. Emphsize the importnce of drwing digrm to represent the sitution. The lnguge in the prolem set will present chllenges for ELL students. Hve ELL students work in pirs, nd give them shorter ssignment so tht they hve time to work through new voculry. Section 9-6: Vector Addition 467

30 Additionl Eplortion Notes Eplortion 9-6 requires students to use compo nents to solve nvigtion prolem. You my need to remind students out the difference etween stndrd-position ngle nd ering nd how to get from one to the other. Allow students 0 minutes to complete this ctivity. Technology Notes Eplortion 9-6 guides students through the ddition of vectors, using properties of tringles. This eplortion cn e done with the id of Sketchpd. CAS Suggestions Vectors re noted on TI-Nspire CAS with squre rckets. As noted in the tet, their components cn e displyed in either component form or using mgnitude nd direction. The two prts of ech vector s definition re seprted y comms even though the output uses spces. Finlly, ngles re indicted using the ngle symol. The first vector from Emple 1 cn e written s [10ft, 70 ] nd the second s [7ft, 10 ]. (Use corresponding ngles to trnsfer the 70 -ngle t the se of the first vector to its tip nd note the 80 ngle is ctully 10 from horizontl.) With this setup, vector ddition gives the mgnitude nd direction of 1. The figure confirms Emple 1. West 70 0 North Bering 0 South 180 Figure 9-6i Est 90 EXAMPLE SOLUTION Nvigtion Prolems A ering, n ngle mesured clockwise from north, is used universlly y nvigtors for velocity or displcement vector. Figure 9-6i shows ering of 0. Victori wlks 90 m due south (ering 180 ), then turns nd wlks 40 m more long ering of 0 (Figure 9-6j).. Find her resultnt displcement vector from the strting point.. Wht is the strting point s ering from the plce where Victori stops? The resultnt vector,. r, goes from the eginning of the first vector to the end of the second. Angle is n ngle in the resulting tringle r (90)(40) cos Use the lw of cosines. r m Store without rounding. To find the ering, first clculte the mesure of ngle in the resulting tringle. cos 9 0 ( ) Use the lw of cosines. (90)( ) Bering r m t ering of These silors continue the trdition of one of the first sefring people. Pcific Islnders red the wves nd clouds to determine currents nd predict wether. 40 m North r 90 m Figure 9-6j See Figure 9-6j. 180 Bering 0 10 ft Chpter 9: 80 7 ft 100 A Using CAS vector nottion, the current in Emple is [4knot, 0 ] nd the ot is [9knot, 130 ]. The vector sum is given in the second line of the net figure with the conversion of the mgnitude ck to knots in line Chpter 9: Tringle Trigonometry

31 Prolem Set 9-6 Reding Anlysis From wht you hve red in this section, wht do you consider to e the min ide? Wht is the difference etween vector quntity nd sclr quntity? Wht is the difference etween vector nd vector quntity? Wht is ment y the unit vectors i nd j, nd how cn they e used to write the components of vector in n y-coordinte system? min Quick Review. The ering from the ending point to the strting point is the opposite of the ering from the strting point to the ending point. To find the opposite, dd 180 to the originl ering. Bering Becuse this ering is greter thn 360, find coterminl ngle y sutrcting 360. Bering Q1. cos 90 A. 1 B. 0 C. 1 D. 1 E. 3 Q. tn 4 A. 1 B. 0 C. 1 D. 1 E. 3 Q3. In FED, the lw of cosines sttes tht f?. Q4. A tringle hs sides ft nd 8 ft nd included ngle 30. Wht is the re of the tringle? Q. For MNO, sin M 0.1, sin N 0.3, nd side m 4 cm. How long is side n? Q6. Finding equtions of two sinusoids tht re comined to form grph is clled?. Q7. If sin 13 nd ngle is in Qudrnt II, wht is cos? Q8. If c s c , then si n 1 (? ). Q9. The eqution y 3 represents prticulr? function. Q10. Wht trnsformtion is pplied to f () to get g() f (3)? For Prolems 1 4, trnslte one vector so tht the two vectors re hed-to-til, nd then use pproprite tringle trigonometry to find nd the ngle the resultnt vector mkes with (Figure 9-6k). Figure 9-6k 1. 7 cm, 11 cm, nd ft, ft, nd in., 0 in., nd mi, 30 mi, nd 1. Displcement Vector Prolem: Lucy wlks on ering of 90 (due est) for 100 m nd then on ering of 180 (due south) for 180 m.. Wht is her ering from the strting point?. Wht is the strting point s ering from where she stops? c. How fr long the ering in prt must Lucy wlk in order to go directly ck to the strting point? Section 9-6: 469 To chnge vectors etween component nd mgnitude-direction forms, one could chnge the system settings or use the conversion commnd. Rect chnges vector to component form, nd Polr chnges vector to mgnitude-direction form. The most vlule prt of these two conversions is tht it doesn t mtter which form the vector ws originlly in. PROBLEM NOTES Supplementry prolems for this section re ville t keyonline. Encourge students to drw digrms to ccompny their work with vectors. Remind them tht vector must include n rrowhed to indicte direction. Using vectors on CAS simplifies much of the lger required when using trigonometric functions. Q1. B Q. A Q3. d e de cos F Q4. 10 ft Q. 60 cm Q6. Hrmonic nlysis Q Q Q9. Eponentil Q10. Horizontl diltion y fctor of 1 3 For Prolems 1 4 nd 6 10, one pproch would e to ssume trvels long the -is. Then the ngle etween the resultnt vector nd the -is is lso the requested ngle etween the resultnt vector nd cm; ft; in.; mi; Prolem nd Prolems 17 0 involve erings rther thn stndrd-position ngles.. Lucy s ering is The strting point s ering from Lucy is c m Section 9-6: Vector Addition 469

32 Prolem Notes (continued) 6. r 1.3 mi/h The resultnt could equl only if the velocities were in the sme direction r 39.3 l; Ae nd Bill neglected the fct tht the mgnitude of the sum of two vectors does not equl the sum of the mgnitudes if the vectors do not point in the sme direction. 8. r km/h; u from the perpendiculr km/h 8c. No. Any upstrem component of your 3 km/h velocity cn never cncel the km/h downstrem component of the wter i j i j i j i j 6. Velocity Vector Prolem: A plne flying with n ir velocity of 400 mi/h crosses the jet strem, which is lowing t 10 mi/h. The ngle etween the two velocity vectors is 4 (Figure 9-6l). The plne s ctul velocity with respect to the ground is the vector sum of these two velocities. 10 mi/h 4 Jet strem s velocity 400 mi/h Figure 9-6l Plne s ir velocity. Wht is the plne s ctul velocity with respect to the ground? Why is it less thn 400 mi/h 10 mi/h?. Wht ngle does the plne s ground velocity vector mke with its 400-mi/h ir velocity vector? 7. Force Vector Prolem: Ae nd Bill cooperte to pull tree stump out of the ground. They think it will tke force of 30 l to do the jo. They tie ropes round the stump. Ae pulls his rope with force of 00 l, nd Bill pulls his rope with force of 10 l. The force vectors mke n ngle of 40, s shown in Figure 9-6m. Stump l 00 l Figure 9-6m Bill Ae. Find the mgnitude of the resultnt force vector nd the ngle the resultnt vector mkes with Ae s vector.. Wht flse ssumption out vectors did Ae nd Bill mke? 8. Swimming Prolem: Suppose tht you swim cross strem tht hs -km/h current.. Find your ctul velocity vector if you swim perpendiculr to the current t 3 km/h.. Find your speed through the wter if you swim perpendiculr to the current ut your resultnt velocity mkes n ngle of 34 with the direction you re heding. c. If you swim t 3 km/h, cn you mke it stright cross the strem? Eplin. For Prolems 9 1, resolve the vector into horizontl nd verticl components v v v 1 v 470 Chpter 9: 470 Chpter 9: Tringle Trigonometry

33 13. Airplne Vector Components Prolem: A jet plne flying with velocity of 00 mi/h through the ir is climing t n ngle of 3 to the horizontl (Figure 9-6n). Verticl component 00 mi/h Velocity 3 vector Horizontl component Figure 9-6n. The mgnitude of the horizontl component of the velocity vector represents the plne s ground speed. Find this ground speed.. The mgnitude of the verticl component of the velocity vector represents the plne s clim rte. How mny feet per second is the plne climing? (Recll tht mile is 80 ft.) 14. Bsell Vector Components Prolem: At time t 0 s, sell is hit with velocity of 10 ft/s t n ngle of to the horizontl (Figure 9 7o). At time t 3 s, the ll hs slowed to 100 ft/s nd is going downwrd t n ngle of 1 to the horizontl. Figure 9-6o. Find the mgnitudes of the horizontl nd verticl components of the velocity vector t time t 0 s. Wht informtion do these components give you out the motion of the sell?. How fst is the sell dropping t time t 3 s? Wht mthemticl quntity revels this informtion? 1. If r 1 units t 70 nd s 40 units t 10, find r s. As sum of two components. As mgnitude nd direction 16. If u 1 units t 60 nd v 8 units t 310, find u v. As sum of two components. As mgnitude nd direction 17. A ship sils 0 mi on ering of 0 nd then turns nd sils 30 mi on ering of 80. Find the resultnt displcement vector s distnce nd ering. 18. A plne flies 30 mi on ering of 00 nd then turns nd flies 40 mi on ering of 10. Find the resultnt displcement vector s distnce nd ering. 19. A plne flies 00 mi/h on ering of 30. The ir is moving with wind speed of 60 mi/h on ering of 190. Find the plne s resultnt velocity vector (speed nd ering) y dding these two velocity vectors. 0. A scu diver swims 100 ft/min on ering of 170. The wter is moving with current of 30 ft/min on ering of 11. Find the diver s resultnt velocity (speed nd ering) y dding these two velocity vectors. 1. Spceship Prolem: A spceship is moving in the plne of the Sun, the Moon, nd Erth. It is eing cted upon y three forces (Figure 9-6p). The Sun pulls with force of 90 newtons t 40. The Moon pulls with force of 0 newtons t 110. Erth pulls with force of 70 newtons t 30. Wht is the resultnt force s sum of two components? Wht is the mgnitude of this force? In wht direction will the spceship move s result of these forces? Moon Erth Sun Figure 9-6p Spceship Section 9-6: horizontl component ; Ground speed is out 410 mi/h. 13. verticl component ; Clim rte 41 ft/s 14. horizontl component ; verticl component The ll is moving with ground speed of out 136 ft/s nd rising t out 63 ft/s. 14. The verticl component of the velocity vector tells the rte t which the ll is dropping. verticl component The ll is dropping t out 1 ft/s. Prolems 1 16 cn e entered s is nd converted to either form s shown in the CAS suggestions i j 1. r units; u i j 16. r units; u r 70 mi t ering of r mi/h t ering of r mi/h t ering of r ft/s t ering of r i j ; r newtons; u Section 9-6: Vector Addition 471

34 Prolem Notes (continued) Prolems eplore the properties of vector ddition. In Prolems nd 6 you my need to remind students wht closure mens. For the set of vectors to e closed under ddition, the sum of ny two vectors must e nother vector... Prolems 6 refer to vectors,, nd c in Figure 9-6q Figure 9-6q. Commuttivity Prolem:. On grph pper, plot y trnslting so tht its til is t the hed of.. On the sme es, plot y trnslting so tht its til is t the hed of. c c. How does your figure show tht vector ddition is commuttive? 3. Associtivity Prolem: Show tht vector ddition is ssocitive y plotting on grph pper ( ) c nd ( c ). 4. Zero Vector Prolem: Plot on grph pper the sum ( ). Wht is the mgnitude of the resultnt vector? Cn you ssign direction to the resultnt vector? Why is the resultnt clled the zero vector?. Closure Under Addition Prolem: How cn you conclude tht the set of vectors is closed under ddition? Why is the eistence of the zero vector necessry to ensure closure? 6. Closure Under Multipliction y Sclr Prolem: How cn you conclude tht the set of vectors is closed under multipliction y sclr? Is the eistence of the zero vector necessry to ensure closure in this cse? 7. Look up the origin of the word sclr. Give the source of your informtion. c. The resultnt vector is the sme regrdless of the order in which you dd the vectors. 3. ( ) c c c c c 4. The mgnitude is 0; the direction is undefined. The resultnt is the vector 0 i 1 0 j, the zero vector. 47 Chpter 9:. If i 1 j nd c i 1 d j re ny two vectors, then,, c, nd d re rel numers. So 1 c nd 1 d re lso rel numers, ecuse the rel numers re closed under ddition. Therefore, the sum ( 1 c) i 1 ( 1 d ) j eists nd is vector, so the set of vectors is closed under ddition. The zero vector is necessry so tht the sum of ny vector i 1 j nd its opposite, i 1 j, will eist. 6. If i 1 j is ny vector, then nd re rel numers. So, if c is ny sclr, i.e., rel numer, then c nd c re rel numers. So the product c i 1 c j eists nd is vector. Therefore, the set of vectors is closed under sclr multipliction. The zero vector is necessry so tht the product of ny vector with the sclr 0 will eist. 7. Sclr is from the Ltin sc le, mening ldder. 47 Chpter 9: Tringle Trigonometry

35 9-7 Ojective Rel-World Tringle Prolems Previously in this chpter you encountered some rel-world tringle prolems in connection with lerning the lw of cosines, the lw of sines, the re formul, nd Hero s formul. You were le to tell which technique to use y the section of the chpter in which the prolem ppered. In this section you will encounter such prolems without hving those eternl clues. Given rel-world prolem, identify tringle nd use the pproprite technique to clculte unknown side lengths nd ngle mesures. To ccomplish this ojective, it helps to formulte some conclusions out which method is pproprite for given set of informtion. Some of these conclusions re contined in this o. Section 9-7 Clss Time dys PLANNING Homework Assignment Dy 1: RA, Q1 Q10, Prolems 1 9 odd Dy : Prolems odd, 18, nd hve students write their own prolems (see Prolem Notes) Surveying instrument PROCEDURES: Tringle Techniques Differentiting Instruction Hve students copy the tringle techniques on pge 473 into their journls nd then rewrite them in their own words. Encourge them to dd digrms to clrify the descriptions. The Reding Anlysis should e done individully nd then checked for ccurcy. Lw of Cosines Usully you use it to find the length of the third side from two sides nd the included ngle (SAS). You cn lso use it in reverse to find n ngle mesure if you know three sides (SSS). You cn use it to find oth lengths of the third side in the miguous SSA cse. Yo u cn t use it if you know only one side ecuse it involves ll three sides. Lw of Sines Usully you use it to find side length when you know n ngle, the opposite side, nd nother ngle (ASA or AAS). You cn lso use it to find n ngle mesure, ut there re two vlues of rcsine etween 0 nd 180 tht could e the nswer. Yo u cn t use it for the SSS cse ecuse you must know t lest one ngle. Yo u cn t use it for the SAS cse ecuse the side opposite the ngle is unknown. Are Formul You cn use it to find the re from two sides nd the included ngle (SAS). Hero s Formul You cn use it to find the re from three sides (SSS). Section 9-7: 473 Hve ELL students work on the prolem set in pirs. Consider shortening the ssignment, nd e prepred to offer support with lnguge. Led Prolem 1 or 13 s whole clss ctivity. Teching Resources Eplortion 9-7: The Ship s Pth Prolem Eplortion 9-7: Are of Regulr Polygon Supplementry Prolems Technology Resources Eplortion 9-7: The Ship s Pth Prolem Clcultor Progrm: AREGPOLY Section Notes TEACHING In this section, students solve vriety of tringle prolems. Some rel-world prolems hve een incorported into erlier sections in this chpter, so two dys is resonle mount of time to spend on this section. Remind students tht in ddition to the new ides from this chpter the lw of sines, lw of cosines, nd vector properties they cn use the right tringle properties they lerned in Chpter. If prolem involves right tringle, it is esier to use property such opposite s sine = thn the lw of sines hypotenuse or lw of cosines. Section 9-7: Rel-World Tringle Prolems 473

36 Section Notes (continued) The procedures o on pge 473 summrizes the tringle techniques students studied in this chpter. Students should copy this informtion into their noteooks, long with the pproprite formuls nd ny pertinent mteril from erlier chpters. This ctivity will help them orgnize the ides in this chpter nd crete guide they cn use when they solve prolems. Consider llowing ech group of students to choose prolem from the prolem set to present to the clss, emphsizing its unique chrcteristics. In ddition to ssigning the prolems in the ook, you might consider sking students to write their own prolems. The prolems students write themselves re often the most interesting nd cretive. Eplortion Notes There re two eplortions for this section. They cn e ssigned in clss or used s group project, group quiz, or n independent homework ssignment. Eplortion 9-7 is rel-world tringle prolem tht involves the miguous cse. From verl description, students re to mke digrm to nlyze the sitution nd solve the prolem. Allow out 0 minutes for this ctivity. Eplortion 9-7 involves the re of regulr polygons nd uses the ide of limit s the numer of sides increses. This eplortion shows numericlly tht the limit of the res of the inscried n-gon pproches the re of the circle ( ) s n increses. Students my need help in writing the progrm in Prolem ; they cn use the progrm AREGPOLY, ville t See the Technology Notes for more informtion out using this progrm. Prolem Set 9-7 Reding Anlysis From wht you hve red in this section, wht do you consider to e the min ide? Under wht condition could you not use the lw of cosines for tringle prolem? Under wht conditions could you not use the lw of sines for tringle prolem? In ech cse tell why you couldn t. min Quick Review Q1. For ABC, write the lw of cosines involving ngle B. Q. For ABC, write the lw of sines involving ngles A nd C. Q3. For ABC, write the re formul involving ngle A. Q4. Sketch XYZ given, y, nd ngle X, showing how you cn drw two possile tringles. Q. Drw sketch showing vector sum. Q6. Drw sketch showing the components of v. Q7. Write if 4 i 7 j nd 6 i 8 j. Q8. cos A. 1 B. 0 C. 1 D. 1 E. 3 Q9. By the composite rgument properties, sin(a B)?. Q10. Wht is the phse displcement of y 7 6 cos ( 37 ) with respect to the prent cosine function? 1. Mountin Height Prolem: A surveying crew hs the jo of mesuring the height of mountin (Figure 9-7). From point on level ground they mesure n ngle of elevtion of 1.6 to the top of the mountin. They move 07 m closer horizontlly nd find tht the ngle of elevtion is now 3.8. How high is the mountin? (You might hve to clculte some other informtion long the wy!) 474 Chpter 9: Technology Notes Eplortion 9-7 in the Instructor s Resource Book sks students to use the properties they ve lerned in this chpter to nswer questions out ship s pth. Students re sked to construct digrm to model the ship s movement, nd this cn esily e done in Sketchpd m Figure 9-7 Height. Studio Prolem: A contrctor plns to uild n rtist s studio with roof tht slopes differently on the two sides (Figure 9-7). On one side, the roof mkes n ngle of 33 with the horizontl. On the other side, which hs window, the roof mkes n ngle of 6 with the horizontl. The wlls of the studio re plnned to e ft prt. Window in roof Wll ft Roof 6 33 Figure 9-7 Wll. Clculte the lengths of the two prts of the roof.. How mny squre feet will need to e pinted for ech tringulr end of the roof? 3. Detour Prolem: Suppose tht you re the pilot of n irliner. You find it necessry to detour round group of thundershowers, s shown in Figure 9-7c. You turn your plne t n ngle of 1 to your originl pth, fly for while, turn, nd then rejoin your originl pth t n ngle of 3, 70 km from where you left it. 1 3 Showers 70 km Figure 9-7c. How much frther did you hve to fly ecuse of the detour? Clcultor Progrm: AREGPOLY sets the clcultor to degree mode, fies nine-digit output, nd then displys res of polygon with rdius 10, s the numer of sides increses. For TI-83 or TI-84, the progrm looks like this: :Degree:Fi 9 :For(X,3,1000) :Disp 0X*sin(360/X) :End 474 Chpter 9: Tringle Trigonometry

37 . Wht is the re of the region enclosed y the tringle? 4. Pumpkin Sle Prolem: Scorpion Gulch Shelter is hving pumpkin sle for Hlloween. The pumpkins will e displyed on tringulr region in the prking lot, with sides 40 ft, 70 ft, nd 100 ft. Ech pumpkin tkes out 3 ft of spce.. Aout how mny pumpkins cn the shelter disply?. Find the mesure of the middle-size ngle.. Underwter Reserch L Prolem: A ship is siling on pth tht will tke it directly over n occupied reserch l on the ocen floor. Initilly, the l is 1000 yd from the ship on line tht mkes n ngle of 6 with the surfce (Figure 9-7d). When the ship s slnt distnce hs decresed to 400 yd, the ship cn contct people in the l y underwter telephone. Find the two distnces from the strting point t which the ship is t slnt distnce of 400 yd from the l. Figure 9-7d 6. Truss Prolem: A uilder hs specifictions for tringulr truss to hold up roof. The horizontl side of the tringle will e 30 ft long. An ngle t one end of this side will e 0. The side to e constructed t the other end will e 0 ft long. Use the lw of sines to find the ngle mesure opposite the 30-ft side. Interpret the results. Q1. 1 c c cos B Q. sin A c sin C Q3. 1 c sin A Q4. Z X y z Y 7. Rocket Prolem: An oserver km from the lunchpd oserves rocket scending verticlly. At one instnt, the ngle of elevtion is 1. Five seconds lter, the ngle hs incresed to 3. Spce shuttle on lunchpd t Cpe Cnverl, Florid. How fr did the rocket trvel during the -s intervl?. Find its verge speed during this intervl. c. If the rocket keeps going verticlly t the sme verge speed, wht will e the ngle of elevtion 1 s fter the first sighting? 8. Grnd Pino Prolem: The lid on grnd pino is held open y 8-in. prop. The se of the prop is in. from the lid s hinges, s shown in Figure 9-7e. At wht possile distnces long the lid could you plce the end of the prop so tht the lid mkes 6 ngle with the pino? Hinges 6 Section 9-7: Q.?? Lid in. Figure 9-7e 8 in. Prop Q7. i 1 j Q8. C 8 in. Q9. sin A cos B cos A sin B Q PROBLEM NOTES Supplementry prolems for this section re ville t keyonline. This prolem set is rrnged so tht odd- nd even-numered prolems re roughly equivlent nd so tht prolems progress from esy to hrd. Aprt from these criteri, there is no prticulr pttern to the rrngement. Students re epected to select the pproprite technique sed on the merits of the prolem. Using CAS to do the lgeric mnipultion for prolems in this section, students will e more confident s they pproch the more difficult prolems, developing criticl thinking skills s they set up equtions nd systems of equtions. Students could use system of equtions to solve Prolem 1. If is the distnce from the se of the mountin s ltitude to the 07 m segment, then h tn 1.6 h 07 1 nd tn CD 44.1 m. Window 1.1 ft; Roof 0.1 ft. Are 10.6 ft km 3. A 607. km pumpkins 4. u yd or yd 6. sin u 30 sin 0 1.1, which is 0 not the sine of ny ngle. It is impossile to uild the truss to the specifictions. The 0-ft side is too short, the 30-ft side is too lrge, or the 0 ngle is too lrge km km/s 7c in. or 3. in. See pge 100 for the nswer to Prolem Q6. Section 9-7: Rel-World Tringle Prolems 47

38 Prolem Notes (continued) km/h km/h 10. u L (l) H (l) The centripetl force is stronger, so the plne is eing forced more strongly wy from stright line into circle. 10c. The horizontl component is 0, so there is no centripetl force to push the plne out of stright pth. 10d e. The plne would strt to fll nd spirl downwrd. 11. Let F the other person s force. F l. Then the mgnitude of the resultnt force vector is out l. 9. Airplne Velocity Prolem: A plne is flying through the ir t speed of 00 km/h. At the sme time, the ir is moving t 40 km/h with respect to the ground t n ngle of 3 with the plne s pth. The plne s ground speed is the mgnitude of the vector sum of the plne s ir velocity nd the wind velocity. Find the plne s ground speed if it is flying. Aginst the wind. With the wind 10. Airplne Lift Prolem: When n irplne is in flight, the ir pressure cretes force vector, clled the lift, tht is perpendiculr to the wings. When the plne nks for turn, this lift vector my e resolved into horizontl nd verticl components. The verticl component hs mgnitude equl to the plne s weight (this is wht holds the plne up). The horizontl component is centripetl force tht mkes the plne go on its curved pth. Suppose tht jet plne weighing 00,000 l nks t n ngle (Figure 9-7f). Figure 9-7f. Mke tle of mgnitudes of lift nd horizontl component for ech from 0 through 30.. Bsed on your tle in prt, why cn plne turn in smller circle when it nks t greter ngle? c. Why does plne fly stright when it is not nking? d. If the mimum lift the wings cn sustin is 600,000 l, wht is the mimum ngle t which the plne cn nk? e. Wht might hppen if the plne tried to nk t n ngle greter thn in prt d? 11. Cnl Brge Prolem: In the pst, it ws common to pull rge with tow ropes on opposite sides of cnl (Figure 9-7g). Assume tht one person eerts force of 0 l t n ngle of 0 with the direction of the cnl. The other person pulls t n ngle of 1 with respect to the cnl with just enough force so tht the resultnt vector is directly long the cnl. Find the force, in pounds, with which the second person must pull nd the mgnitude of the resultnt force vector. Brge 0 1 Figure 9-7g Cnl 1. Silot Force Vector Prolem: Figure 9-7h represents silot with one sil, set t 30 ngle with the is of the ot. The wind eerts force vector of 300 l tht cts on the mst in direction perpendiculr to the sil. Sil 30 Figure 9-7h 300 l Force vector. Find the solute vlue of the component of the force vector long the is of the ot. (This force mkes the ot move forwrd.). How hrd is the wind pushing the ot in the direction perpendiculr to the is of the ot? (The keel minimizes the effect of this force in pushing the ot sidewys.) c. On the Internet or in some other reference source, look up the physics of silots to find out why two sils more thn doule the forwrd force produced y one sil. Give the source of your informtion. 476 Chpter 9: 476 Chpter 9: Tringle Trigonometry

39 Figure 9-7i 13. Truck on Hill Prolem: One of the steepest streets in the United Sttes is Mrin Street, in Berkeley, Cliforni. In some locks the street mkes 13 ngle with the horizontl. Suppose tht truck is prked on such street (Figure 9-7i). The 40,000-l weight vector of the truck cn e resolved into components perpendiculr to the street surfce (the norml component) nd prllel to the street surfce. Prllel component 13 40,000 l Figure 9-7i Mrin Street Norml component. Find the mgnitude of the norml component. Show tht it is not much less thn the 40,000-l weight of the truck.. Find the mgnitude of the prllel component. Is this surprising? This is the force the rkes must eert to keep the truck from rolling down the hill. 14. Sliding Friction Force Prolem: Figure 9-7j, left, shows 100-l o eing pulled cross level floor. Figure 9-7j, right, shows the sme o eing pulled up rmp (n inclined plne, s physicists cll it) tht mkes 7 ngle with the horizontl. In this prolem you will lern how to clculte the force needed to pull the o up the rmp. Friction Prllel Pull force Pull? component Friction force Friction Inclined force Floor (level) plne (Rmp) Norml Weight: 100 l component 7 Weight: 100 l Figure 9-7j. To pull the o long the level floor, ll you need to do is overcome the force of friction. The mgnitude of this friction force is directly proportionl to the mgnitude of the force cting perpendiculr to the floor (the norml force), in this cse the weight of the o. Suppose tht it tkes 60-l force to pull 100-l o cross the floor. Let e the mgnitude of the friction force, nd let y e the mgnitude of the norml force. Write the prticulr eqution epressing y s function of. (The proportionlity constnt in this eqution is clled the coefficient of friction.). When the o is eing pulled up the rmp, the norml force is the component of the weight in the direction perpendiculr to the rmp (Figure 9-7j, right). Assuming tht the coefficient of friction for the rmp is the sme s for the floor, use your eqution from prt to clculte the mgnitude, y, of the force needed to overcome friction for the 100-l o. c. The totl force needed to move the o up the rmp is the sum of the force needed to overcome friction (prt ) nd the component of the weight prllel to the rmp. How hrd must you pull on the o to move it up the rmp? d. On the Internet or in nother reference source, find the difference etween sttic coefficient of friction nd dynmic coefficient of friction. Give the source of your informtion. 1. il component 10 l 1. norml component l 1c. Answers will vry, ut the mjor effect is the increse in wind speed s the wind goes through the reltively nrrow spce etween the sils, creting thrust vector tht hs n dditionl component in the il direction. 13. norml component 38, ,000 l, which is not much less thn the weight of the truck. 13. prllel component l, which is surprisingly lrge! 14. y y l 14c. Totl force l 14d. Answers will vry, ut the mjor difference is tht the coefficient of sttic friction is used to clculte the force necessry to strt sttionry oject moving, wheres the coefficient of dynmic friction, usully smller, is used to clculte the force needed to keep n oject in motion once it hs een strted. Section 9-7: 477 Section 9-7: Rel-World Tringle Prolems 477

40 Prolem Notes (continued) l; resultnt force l 1. u tn 1 10 The grph shows tht u pproches horizontl symptote t 90 s gets lrger c. String tension resultnt force 10 1 The grph shows tht the tension pproches symptoticlly s gets lrger Tension t 1 cos 0 t cos 40 t 1 cos 0 t cos t 1 sin 0 1 t sin c. t l; t l 16d. t 1 cos cos l; t cos cos l The two horizontl components re equl. t 1 sin 0 1 t sin sin sin 40 0 The two verticl components sum to 0. 16e. t ers more thn twice the mount of the 0-l weight s t t ; t , 1. Hnging Weight Prolem 1: Figure 9-7k shows 10-l weight hnging on string 0 in. long. You pull the weight sidewys with force of mgnitude, in pounds, mking the string form n ngle with the verticl. In this prolem you will find the mesure of ngle s function of how hrd you pull nd the resulting tension force in the string. 10 l 0 in. 478 Chpter 9: Figure 9-7k Pull Resultnt force. The resultnt force eerted on the string y the lock is the vector sum of the 10-l weight of the lock nd the -l force, nd it cts in the direction of the string. With wht force must you pull to mke 30? Wht will e the tension in the string (the mgnitude of the resultnt vector)?. Write n eqution epressing s function of. Sketch the grph of this function. Wht hppens to the ngle mesure s ecomes very lrge? c. Write nother eqution epressing the tension in the string s function of. Sketch the grph of this function. Wht hppens to this tension s ecomes very lrge? 16. Hnging Weight Prolem : Figure 9-7l shows n oject weighing 0 l supported y two cles connected to wlls 6 ft prt on opposite sides of n lley. Tension vectors t 1 nd t in the cles mke ngles of 0 nd 40, respectively, with the horizontl. The resultnt vector of these tension vectors is the 0-l vector pointed stright up, in direction opposite to the weight vector. In this prolem you will clculte the mgnitudes of the two tension vectors. t Figure 9-7l t. The horizontl components of vectors t 1 nd t hve opposite directions ut equl mgnitudes. (Otherwise the oject would move sidewys!) Write n eqution involving these mgnitudes tht epresses this fct.. The verticl components of t 1 nd t sum to the upwrd-pointing 0-l vector. Write nother eqution involving the mgnitudes of these tension vectors tht epresses this fct. c. Solve the system of equtions in prts nd to find the mgnitudes of t 1 nd t. Store the results without rounding. d. Demonstrte numericlly tht the mgnitudes of the horizontl components of t 1 nd t re equl nd tht the mgnitudes of the verticl components sum to 0 l. e. Which tension vector ers more of the 0-l weight, the one with the lrger ngle to the horizontl or the one with the smller ngle? 17. Hnging Weight y Lw of Sines Prolem: Figure 9-7m shows the two tension vectors t 1 nd t from Figure 9-7l drwn hed-to-til, with the 0-l sum vector strting t the til of t 1 nd ending t the hed of t.use the lw of sines to find the mgnitudes of t 1 nd t. t t Figure 9-7m 0 l 478 Chpter 9: Tringle Trigonometry

41 18. Ship s Velocity Prolem: A ship is siling through the wter in the English Chnnel with velocity knots on ering of 17, s shown in Figure 9-7n. The current hs velocity knots on ering of 13. The ctul velocity of the ship is the vector sum of the ship s velocity nd the current s velocity. Find the ship s ctul velocity. knots 13 North 17 knots c. Severl orits lter, only Stellite 1 is visile, while Stellite is ner the opposite side of Erth (Figure 9-7o). Ivn determines tht the mesure of ngle A is 37.7, the mesure of ngle B is 113, nd the distnce etween him nd Stellite 1 is 436 km. To the nerest kilometer, how fr prt re Ivn nd Stellite? Stellite B Stellite 1 Figure 9-7o Ivn A 436 km 18. r knots t ering of r km/h t ering of km 0. The lrgest ngle is t the spce sttion; u ; Are 16. km 0c. 80 km km Figure 9-7n 19. Wind Velocity Prolem: A nvigtor on n irplne knows tht the plne s velocity through the ir is 0 km/h on ering of 37. By oserving the motion of the plne s shdow cross the ground, she finds to her surprise tht the plne s ground speed is only km/h nd tht its direction is long ering of 1. She relizes tht the ground velocity is the vector sum of the plne s velocity nd the wind velocity. Wht wind velocity would ccount for the oserved ground velocity? 0. Spce Sttion Prolem: Ivn is in spce sttion oriting Erth. He hs the jo of oserving the motion of two communictions stellites.. As Ivn pproches the two stellites, he finds tht one of them is 8 km wy, the other is 11 km wy, nd the ngle etween the two (with Ivn t the verte) is 10. How fr prt re the stellites?. A few minutes lter, Stellite 1 is km from Ivn nd Stellite is 7 km from him. At this time, the two stellites re 10 km prt. At which of the three spce vehicles does the lrgest ngle of the resulting tringle occur? Wht is the mesure of this ngle? Wht is the re of the tringle? The Interntionl Spce Sttion is joint project of the United Sttes, the Russin Federtion, Jpn, the Europen Union, Cnd, nd Brzil. Construction egn in 1998 nd continues tody through the efforts of stronuts who live ord the sttion for mny months t time. 1. Visiility Prolem: Suppose tht you re ord plne destined for Hwii. The pilot nnounces tht your ltitude is 10 km. You decide to clculte how fr wy the horizon is. You drw sketch s in Figure 9-7p nd relize tht you must clculte n rc length. You recll from geogrphy tht the rdius of Erth is out 6400 km. How fr wy is the horizon long Erth s curved surfce? Is this surprising? How fr? 10 km You Horizon Figure 9-7p Section 9-7: 479 Section 9-7: Rel-World Tringle Prolems 479

42 Prolem Notes (continued) cm or cm. (00 cos 0 ) 4? 1? , so there is no possile solution. Or note tht when u 0, the height of the hinge is 100 sin cm, which is greter thn the length of the second ruler. c Are m m m 4c ; 8.0. Answers will vry.. The progrm should give the epected nswer. c. Lel the 9 ngle A, nd lel the rest of the vertices clockwise s B through F. AC 0 1? 0? cos m ACB sin 1 0 sin AC ACD 147 ACB AD AC 1 1? AC? 1 cos ACD m ADC sin 1 AC sin ACD AD ADE 1 ADC AE AD 1 18? AD? 18 cos ADE m AED sin 1 AD sin ADE AE AEF 11 AED AF AE 1 17? AE? 17 cos AEF m. Hinged Rulers Prolem: Figure 9-7q shows meterstick (100-cm ruler) with 60-cm ruler ttched to one end y hinge. The other ends of oth rulers rest on horizontl surfce. The hinge is pulled upwrd so tht the meterstick mkes n ngle with the surfce. 100 cm 480 Chpter 9: How long? Hinge Figure 9-7q 60 cm. Find the two possile distnces etween the ruler ends if 0.. Show tht there is no possile tringle if 0. c. Find the vlue of tht gives just one possile distnce etween the ends. 3. Surveying Prolem 1: A surveyor mesures the three sides of tringulr field nd gets lengths 114 m, 16 m, nd 7 m.. Wht is the mesure of the lrgest ngle of the tringle?. Wht is the re of the field? 4. Surveying Prolem : A field hs the shpe of qudrilterl tht is not rectngle. Three sides mesure 0 m, 60 m, nd 70 m, nd two ngles mesure 17 nd 13 (Figure 9-7r). 60 m 0 m Figure 9-7r 70 m. By dividing the qudrilterl into two tringles, find its re.. Find the length of the fourth side. c. Find the mesures of the other two ngles. d. For nonconve polygon, you might not e le to divide it into tringles tht fn out rdilly from single verte.. Surveying Prolem 3: Surveyors find the re of n irregulrly shped trct of lnd y tking field notes. These notes consist of the length of ech side nd informtion for finding ech ngle mesure. For this prolem, strting t one verte, the trct is divided into tringles. For the first tringle, two sides nd the included ngle re known (Figure 9-7s), so you cn clculte its re. To clculte the re of the net tringle, you must recognize tht one of its sides is lso the third side of the fi r s t tringle nd tht one of its ngles is n ngle of the polygon (147 in Figure 9-7s) minus n ngle of the first tringle. By clculting the mesures of this side nd ngle nd using the net side of the polygon (1 m in Figure 9-7s), you cn clculte the re of the second tringle. The res of the remining tringles re clculted in the sme mnner. The re of the trct is the sum of the res of the tringles. 0 m 9 m m Figure 9-7s 4 1 m m 17 m. Write progrm for clculting the re of trct using the technique descried. The input should e the mesures of the sides nd ngles of the polygon. The output should e the re of the trct.. Use your progrm to clculte the re of the trct in Figure 9-7s. If you get pproimtely m, you cn ssume tht your progrm is working correctly. c. Show tht the lst side of the polygon hs length m, which is close to the mesured vlue, 31 m. d. The polygon in Figure 9-7s is conve polygon ecuse none of the ngles mesure more thn 180. Eplin why your progrm might give wrong nswers if the polygon were not conve. 480 Chpter 9: Tringle Trigonometry

43 9-8 Review Prolems Chpter Review nd Test In this chpter you returned to the nlysis of tringles strted in Chpter. You epnded your knowledge of trigonometry to include olique tringles s well s right tringles. You lerned techniques to find side lengths nd ngle mesures for vrious sets of given informtion. These techniques re useful for rel-world prolems, including nlyzing vectors. R0. Updte your journl with things you lerned in Plot the dt from prts nd nd this this chpter. Include topics such s the lws of eqution for y on the sme screen. Do cosines nd sines, the re formuls, how these the dt seem to fit the lw of cosines? re derived, nd when it is pproprite to use Does the grph seem to e prt of them. Also include how tringle trigonometry sinusoid? Eplin. is pplied to vectors. R.. Mke sketch of tringle with sides 0 ft R1. Figure 9-8 shows tringles with sides 4 cm nd 30 ft nd included ngle 13. Find the nd cm, with vrying included ngle. The length of the third side. length of the third side (dshed) is function of. Mke sketch of tringle with sides 8 m,. The five vlues of shown re 30, 60, 90, m, nd 11 m. Clculte the mesure of the 10, nd 10. lrgest ngle.. Mesure the length of the third side (dshed) c. Suppose you wnt to construct tringle for ech tringle. with sides 3 cm, cm, nd 10 cm. Eplin. How long would the third side e if the ngle why this is geometriclly impossile. Show were 180? If it were 0? how computtion of n ngle using the lw c. If 90, you cn clculte the length of of cosines leds to the sme conclusion. the dshed line y mens of the Pythgoren d. Sketch DEF with ngle D in stndrd theorem. Does your mesured length in position in uv-coordinte system. Find the prt gree with this clculted length? coordintes of points E nd F in terms of d. If y is the length of the dshed line, the lw of sides e nd f nd ngle D. Use the distnce cosines sttes tht formul to prove tht you cn clculte d using y 4 4 cos d e f ef cos D v 4 0 Figure u Section 9-8 PLANNING Clss Time dys (including 1 dy for testing) Homework Assignment Dy 1: R0 R7, T1 T1 Dy (fter Chpter 9 Test): Begin the Cumultive Review (Section 9-9), Prolems 1 18 Teching Resources Blckline Mster Prolem T1 Supplementry Prolems Test, Chpter 9, Forms A nd B Section Notes TEACHING Section 9-8 contins set of review prolems, set of concept prolems, nd chpter test. The review prolems include one prolem for ech section in the chpter. You my wish to use the chpter test s n dditionl set of review prolems. Encourge students to prctice the no-clcultor prolems without clcultor so tht they re prepred for the test prolems for which they cnnot use clcultor. R0. Journl entries will vry. R1. Answers my vry slightly. u Third Side (cm) Section 9-8: R ; 4 1 R1c ; Yes R1d Third side (cm) No, the shpe is not sinusoid. 481 See pge 100 for the nswers to Prolem R. Section 9-8: Chpter Review nd Test 481

44 Differentiting Instruction Students should do the review prolems in pirs. Go over the review prolems in clss, perhps y hving students present their solutions. You might ssign students to write up their solutions efore clss strts. Work through the concept prolems s clss ctivity to give students nother opportunity to mster the new voculry. For Prolem C, eplin dot product nd mention tht there eists nother type of vector product clled the cross product. Model good eplntions for Prolems T4 T7, ut tke lnguge difficulties into ccount when ssessing student responses. Encourge students to use digrms s prt of their eplntions. Even with lnguge support, the cumultive review will proly tke too much time for ELL students to complete. Allow students to work in pirs, nd shorten the ssignment. Becuse mny cultures norms highly vlue helping peers, ELL students often help ech other on tests. You cn limit this tendency y mking multiple versions of the test. Consider giving group test the dy efore the individul test, so tht students cn lern from ech other s they review, nd they cn identify wht they don t know prior to the individul test. Give copy of the test to ech group memer, hve them work together, then rndomly choose one pper from the group to grde. Grde the test on the spot, so students know wht they need to review further. Mke this test worth 1 3 the vlue of the individul test, or less. ELL students my need more time to tke the test. ELL students will enefit from hving ccess to their ilingul dictionries while tking the test. R3.. Mke sketch of tringle with sides 0 ft nd 30 ft nd included ngle 13. Find the re of the tringle.. Mke sketch of tringle with sides 8 mi, 11 mi, nd 1 mi. Find the mesure of one ngle nd use it to find the re of the tringle. Clculte the re gin using Hero s formul. Show tht the results re the sme. c. Suppose tht two sides of tringle hve lengths 10 yd nd 1 yd nd tht the re is 40 yd. Find the two possile mesures of the included ngle etween these two sides. d. Sketch DEF with side d horizontl. Drw the ltitude from verte D to side d. Wht does this ltitude equl in terms of side e nd ngle F? By pproprite geometry, show tht the re of the tringle is Are 1 de sin F R4.. Mke sketch of tringle with one side 6 in., the ngle opposite tht side 39, nd nother ngle, 48. Clculte the length of the side opposite the 48 ngle.. Mke sketch of tringle with one side m nd its two djcent ngles mesuring 11 nd 38. Find the length of the longest side of the tringle. c. Mke sketch of tringle with one side 7 cm, second side cm, nd the ngle opposite the -cm side 31. Find the two possile mesures of the ngle opposite the 7-cm side. d. Sketch DEF nd show sides d, e, nd f. Write the re three wys: in terms of ngle D, in terms of ngle E, nd in terms of ngle F. Equte the res nd then perform clcultions to derive the three-prt eqution epressing the lw of sines. R. Figure 9-8 shows tringle with sides cm nd 8 cm nd ngles nd, not included y these sides. 8 cm 48 Chpter 9: Figure 9-8 cm. If, clculte the two possile vlues of the length of the third side.. If 8, show lgericlly tht there is no possile tringle. c. Clculte the vlue of for which there is ectly one possile tringle. d. If 47, clculte the one possile length of the third side of the tringle. R6.. Vectors nd mke 174 ngle when plced til-to-til (Figure 9-8c). The mgnitudes of the vectors re 6 nd 10. Find the mgnitude of the resultnt vector nd the ngle this resultnt vector mkes with when they re plced til-to-til Figure 9-8c. Suppose tht i 3 j nd 7 i 6 j. Find the resultnt vector s sums of components. Then find the vector gin s mgnitude nd n ngle in stndrd position. c. A ship moves west (ering of 70 ) for 10 mi nd then turns nd moves on ering of 130 for nother 00 mi. How fr is the ship from its strting point? Wht is the ship s ering reltive to its strting point? d. A plne flies through the ir t 300 km/h on ering of 0. Menwhile, the ir is moving t 60 km/h on ering of 11. Find the plne s resultnt ground velocity s sum of two components, where unit vector i points north nd j points est. Then find the plne s resultnt ground speed nd the ering on which it is ctully moving. 48 Chpter 9: Tringle Trigonometry

45 e. Clvin s Roof Vector Prolem: Clvin does roof repirs. Figure 9-8d shows him sitting on roof tht mkes n ngle with the horizontl. The prllel component of his 160-l weight vector cts to pull him down the roof. The frictionl force vector countercts the prllel component with mgnitude (Greek letter mu) times the mgnitude of the norml component of the force vector. Here is the coefficient of friction, nonnegtive constnt which is usully less thn or equl to 1. If 0.9 nd 40, will Clvin e le to sit on the roof without sliding? Wht is the steepest roof Clvin cn sit on without sliding? Why could Clvin never e held y friction lone on roof with 4? Clvin Prllel component 160 l Roof Figure 9-8d Friction force Norml component R7. Airport Prolem (prts f): Figure 9-8e shows Ngoy Airport nd Tokyo Airport 60 km prt. The ground controllers t Tokyo Airport monitor plnes within 100-km rdius of the irport.. Plne 1 is 0 km from Ngoy Airport t n ngle of 3 to the stright line etween the irports. How fr is Plne 1 from Tokyo Airport? Is it relly out of rnge of Tokyo Ground Control, s suggested y Figure 9-8e? Ngoy Airport 0 km 3 60 km Plne 3 Plne 1 First Figure 9-8e R3. A 340. ft 30 R Tokyo Airport Plne s pth Lst 100 km u ; A mi ; s 17; A mi. Plne is going to tke off from Ngoy Airport nd fly pst Tokyo Airport. Its pth will mke n ngle with the line etween the irports. If 1, how fr will Plne e from Ngoy Airport when it first comes within rnge of Tokyo Ground Control? How fr from Ngoy Airport is it when it is lst within rnge? Store oth of these distnces in your clcultor, without rounding. c. Show tht if 40, Plne is never within rnge of Tokyo Ground Control. d. Clculte the vlue of for which Plne is within rnge of Tokyo Ground Control t just one point. How fr from Ngoy Airport is this point? Store the distnce in your clcultor, without rounding. e. Show numericlly tht the squre of the distnce in prt d is ectly equl to the product of the two distnces in prt. Wht theorem from geometry epresses this result? f. Plne 3 (Figure 9-8e) reports tht it is eing forced to lnd on n islnd t se! Ngoy Airport nd Tokyo Airport report tht the ngle mesures etween Plne 3 s position nd the line etween the irports re 3 nd 7, respectively. Which irport is Plne 3 closer to? How much closer? Section 9-8: R3c. u 41.8 or 138. R3d. F e D d 483 ltitude e sin F ; A 1 h ; = 1 de sin F R cm or 3.4 cm f E PROBLEM NOTES Supplementry prolems for this section re ville t keyonline. R. sin 1.6, which is not the sine of ny ngle. Rc. The -cm side must e perpendiculr to the third side, mking the 8-cm side the hypotenuse of right 1 tringle. Then u sin Rd. 10. cm R6. r ; 16. R i 3 j ; r 1.4 ; u R6c. r mi t ering of R6d. r i j ; r = t ering of R6e. Clvin will not slide down ecuse the friction force is greter thn the mgnitude of the prllel component. The steepest ngle is it less thn 4. If u 4, then the prllel component hs mgnitude greter thn tht of the norml component, so friction lone could not keep Clvin from sliding down the roof. R km, so it is out of rnge. R km or km R7c. (0 cos 40 ) 4? 1? 7,600 71, , so is undefined. R7d. The line from the plne to Tokyo Airport must e perpendiculr to the flight pth, so.6 R7e. 40 7,600 ( )( ) The theorem sttes tht if P is point eterior to circle C, PR cuts C t Q nd R, nd PS is tngent to C t S, then PQ? PR PS. R7f. Ngoy Airport is closer y out 3. km. See pge 100 for the nswers to Prolem R4. Section 9-8: Chpter Review nd Test 483

46 Prolem Notes (continued) R7g R7h l R7i. The helicopter cn tilt so tht the thrust vector ectly cncels the wind vector. C1. Student essy C ? 6? 7 cos ft 6 1 7? 6? 7 cos ft The nswers re the sme ecuse cos 0 cos 110. C. 1? 6? 7 sin ft 1? 6? 7 sin ft The nswers re opposite ecuse sin 0 sin 110. Cc. A nbcd ft ; A ABCD 30.7 ft Directly: First find /C. DB 6 1 7? 6? 7 cos ? 10? 1 cos C 40 cos C cos 0 C cos cos ? 10? 1 sin C ft C3. Student project? 6? 7 sin 0 Helicopter Prolem (prts g i): The rotor on helicopter cretes n upwrd force vector (Figure 9-8f). The verticl component of this force (the lift) lnces the weight of the helicopter nd keeps it in the ir. The horizontl component (the thrust) mkes the helicopter move forwrd. Suppose tht the helicopter weighs 3000 l. g. At wht ngle will the helicopter hve to tilt forwrd to crete thrust of 400 l? h. Wht will e the mgnitude of the totl force vector? Concept Prolems C1. Essy Project: Reserch the contriutions of different cultures to trigonometry. Use these resources or others you might find on the We or in your locl lirry: Eli Mor, Trigonometric Delights (Princeton: Princeton University Press, 1998); Dvid Bltner, The Joy of (New York: Wlker Pulishing Co., 1997). Write n essy out wht you hve lerned. C. Refle Angle Prolem: Figure 9-8g shows qudrilterl ABCD, in which ngle A is refle ngle mesuring 0. The resulting figure is clled nonconve polygon. Note tht the digonl from verte B to D lies outside the figure. D C A Figure 9-8g B i. Eplin why the helicopter cn hover over the sme spot y judicious choice of the tilt ngle. Figure 9-8f. Find the mesure of ngle A in ABD. Net, clculte the length DB using the side lengths 6 ft nd 7 ft shown in Figure 9-8g. Then clculte DB directly, using the 0 mesure of ngle A. Do you get the sme nswer? Eplin why or why not.. Clculte the re of ABD using the nonrefle ngle you clculted in prt. Then clculte the re of this tringle directly using the 0 mesure of ngle A. Do you get the sme nswer for the re? Eplin why or why not. c. Use the results in prt to find the re of BCD. Then find the re of qudrilterl ABCD. Eplin how you cn find this re directly using the 0 mesure of ngle A. C3. Angle of Elevtion Eperiment: Construct n inclinometer tht you cn use to mesure ngles of elevtion. One wy to do this is to hng piece of wire, such s strightened pper clip, from the hole in protrctor, s shown in Figure 9-8h. Then tpe strw to the protrctor so tht you cn sight distnt oject more ccurtely. As you view the top of uilding or tree long the stright edge of the protrctor, grvity holds the pper clip verticl, llowing you to determine the ngle of elevtion. 484 Chpter 9: 484 Chpter 9: Tringle Trigonometry

47 Use your pprtus to mesure the height of tree or uilding using the techniques of this chpter. Sod strw Protrctor Inclinometer Figure 9-8h Wire C4. Euclid s Prolem: This prolem comes from Euclid s Elements. Figure 9-8i shows circle with secnt line nd tngent line. P Tngent Q S Secnt Figure 9-8i. Sketch similr figure using dynmic geometry progrm, such s The Geometer s Sketchpd, nd mesure the lengths of the secnt segments, PQ nd PR, nd the tngent segment PS. By vrying the rdius of the circle nd the ngle QPO, see if it is true tht PS PQ PR. Using the trigonometric lws nd identities you ve lerned, prove tht the eqution in prt is true sttement. O R Euclid of Alendri C. Dot (Sclr) Product of Two Vectors Prolem: Figure 9-8j shows two vectors in stndrd position: 3 i 4 j 7 i j The dot product, written to e cos, is defined where is the ngle etween the two vectors when they re plced til-to-til. Find the mesure of the ngle etween nd, nd store it without rounding. Use the result nd the ect lengths of nd to clculte. You should find tht the nswer is n integer! Figure out wy to clculte using only the coefficients of the unit vectors: 3, 4, 7, nd. Why do you suppose the dot product is lso clled the sclr product of the two vectors? C4. Sketch should mtch Figure 9-8i. C4. Becuse ech is rdius of the circle, let SO, QO, RO r. By the Pythgoren property, (1) PO PS 1 r By the lw of cosines, () r PQ 1 PO (PQ )(PO) cos (3) r PR 1 PO (PR)(PO) cos Sustituting (1) into () nd (3) nd rerrnging, (4) (PQ )(PO) cos PQ 1 PS () (PR)(PO) cos PR 1 PS Dividing () into (4), PQ PR PQ 1 PS PR 1 PS By multiplying y common denomintor, rerrnging, nd fctoring, PQ? PR 1 PQ? PS PR? PQ 1 PR? PS PQ? PR PR? PQ PR? PS PQ? PS PQ? PR(PR PQ ) PS (PR PQ ) PQ? PR PS C. u ;? 9 The dot product cn lso e clculted y finding the sum of the products of the i coefficients nd the j coefficients:? 3? 7 1 4? 9. This method is covered in Chpter 1. The dot product is clled the sclr product ecuse the nswer is sclr, not vector. Figure 9-8j 10 Section 9-8: 48 Section 9-8: Chpter Review nd Test 48

48 Prolem Notes (continued) T1. d c 1 e ce cos D T. c sin C d sin D e sin E or sin c C sin D d sin e E T3. A 1 de sin C T4. ASA is shown, ut the lw of cosines works only for SAS nd SSA. T. SAS is shown, ut the lw of sines works only for ASA, SAA, nd SSA. T T7. The rnge of cos 1 is 0 # u # 180, which includes every possile ngle mesure for tringle. But the rnge of sin 1 is 90 # u # 90, so the function sin 1 cnnot find otuse ngles. T8. T9. j y 3i 3 3i j T10. Student drwing. The third side should e out 3. cm. T cm T1. Chpter Test Prt 1: No clcultors (T1 T9) T9) T7. Eplin why you cn use the inverse cosine function, cos 1, when you re finding n ngle To nswer Prolems T1 T3, refer to Figure 9-8k. of tringle y the lw of cosines ut must C use the inverse sine reltion, rcsin, when you d re finding n ngle of tringle y the lw of E sines. e c T8. Sketch the vector sum (Figure 9-8o). D Figure 9-8k T1. Write the lw of cosines involving ngle D. T. Write the lw of sines (either form). T3. Write the re formul involving sides d nd e. T4. Eplin why you cnnot use the lw of cosines for the tringle in Figure 9-8l cm Figure 9-8l 0 T. Eplin why you cnnot use the lw of sines for the tringle in Figure 9-8m. 4 cm cm Figure 9-8m T6. Eplin why there is no tringle with the side lengths given in Figure 9-8n. 10 cm 19 cm Figure 9-8n 7 cm Figure 9-8o T9. Sketch vector v 3 i j nd its components in the - nd y-directions. Prt : Grphing clcultors re llowed (T10 T) T10. Construct tringle with sides 7 cm nd cm nd n included ngle 4. Mesure the third side. T11. Clculte the length of the third side in Prolem T10. Does the mesurement in Prolem T10 gree with this clculted vlue? T1. Mke sketch of tringle with se 0 ft nd se ngles 38 nd 47. Clculte the mesure of the third ngle. T13. Clculte the length of the shortest side of the tringle in Prolem T1. T14. Sketch tringle. Mke up lengths for the three sides tht give possile tringle. Clculte the mesure of the lrgest ngle. Store the nswer without rounding. T1. Find the re of the tringle in Prolem T14. Use the ngle mesure you clculted in Prolem T14. Store the nswer without rounding. T16. Use Hero s formul to clculte the re of your tringle in Prolem T14. Does it gree with your nswer to Prolem T1? Chpter 9: The third ngle mesures Chpter 9: Tringle Trigonometry

49 T17. Figure 9-8p shows circle of rdius 3 cm. Point P is cm from the center. From point P, secnt line is drwn t n ngle of 6 to the line connecting the center to P. Use the lw of cosines to clculte the two unknown lengths leled nd in the figure. P 6 Tngent cm 3 cm 3 cm Figure 9-8p 3 cm T18. Recll tht the rdius of circle drwn to the point of tngency is perpendiculr to the tngent. Use this fct to clculte the length of the tngent segment from point P in Figure 9-8p. T19. Show numericlly tht the product of the two lengths you found in Prolem T17 equls the squre of the tngent length you found in Prolem T18. This geometricl property ppers in Euclid s Elements. T0. For v 3 i j, clculte the mgnitude. Clculte the direction s n ngle in stndrd position. T1. Vector Difference Prolem: Figure 9-8q shows position vectors 3 i 4 j 7 i j By sutrcting components, find the difference vector, d. On copy of Figure 9-8q, show tht d is equl to the displcement vector from the hed of to the hed of. Eplin how this interprettion of vector difference is nlogous to the wy you determine how fr your cr hs gone y sutrcting the eginning odometer reding from the ending odometer reding. Figure 9-8q T. Wht did you lern s result of tking this test tht you did not know efore? 10 T ft T14 T16. Answers will vry. T cm or cm T18. 4 cm T19. ( )( ) 16 4 T0. r.8; u A lckline mster for Prolem T1 is ville in the Instructor's Resource Book. T1. 4 i 1 j The grph shows tht d equls the displcement from the hed of to the hed of, nlogous to where you end minus where you egn. u T. Answers will vry. d 10 Section 9-8: 487 Section 9-8: Chpter Review nd Test 487

50 Section 9-9 PLANNING 9-9 Cumultive Review, Chpters 9 These prolems constitute - to 3-hour rehersl for your emintion on the trigonometric functions unit, Chpters 9. You egn y studying periodic functions. Clss Time 1 or dys Homework Assignment Dy 1: Keep working on the Cumultive Review, Prolems Dy : Complete the Cumultive Review, Prolems 38 46, nd do Prolem Set 10-1 Teching Resources Blckline Mster Prolem 4d Test 6, Cumultive Test, Chpters 9, Forms A nd B Section Notes TEACHING The cumultive review questions in this section will help students reherse for n em on the trigonometric functions unit. Whenever possile, the prolems re pplied to rel-world situtions. A cumultive em cn e quite n ordel for students. Students working in smll groups cn hve fun with these cumultive review prolems nd lern lot from ech other. Students should lso e encourged to look over their old tests nd quizzes nd ring in ny prolems they still don t understnd. You my lso wnt to mke up n dditionl set of prctice prolems tht complement this prolem set. Students should consult their journls to id in the review process. If you re giving cumultive test, use this prolem set s guide for the type of prolems to include. Use your judgment out the kind of review you will provide nd the kind of cumultive em you will give your students. Review Prolems 1. Stellite Prolem 1: A stellite is in circulr orit 7. Wht specil nme is given to the kind of round Erth. From where you re on Erth s periodic function you grphed in Prolem 6? surfce, the stright-line distnce to the stellite Periodic functions such s the one in Prolem 1 hve (through Erth, t times) is periodic function of time. Sketch resonle grph. independent vriles tht cn e time or distnce, not n ngle mesure. So you lerned out circulr functions whose independent vrile is, not. The rdin is the link etween trigonometric functions nd circulr functions. 8. How mny rdins re in 360? 180? 90? 4? 9. How mny degrees re in rdins? To write equtions for periodic functions such s the one in Prolem 1, you generlized the trigonometric functions from geometry y llowing ngles to e negtive or greter thn Sketch 13 ngle in stndrd position. Drw the reference tringle nd find the mesure of the reference ngle. 3. The terminl side of ngle contins the point (1, ) in the uv-coordinte system. Write the ect vlues (no decimls) of the si trigonometric functions of. 4. Write the ect vlue (no decimls) of sin 40.. Drw 180 in stndrd position. Eplin why cos If is llowed to tke on ny rel numer of degrees, the trigonometric functions ecome periodic functions of. 6. Sketch the grph of the prent sine function, y sin. 488 Chpter 9: d 1.. t 10. Sketch grph showing the unit circle centered t the origin of uv-coordinte system. Sketch n -is tngent to the circle, going verticlly through the point (u, v) (1, 0). If the -is is wrpped round the unit circle, show tht the point (, 0) on the -is corresponds to ngle mesure rdins. 11. Sketch the grph of the prent circulr sinusoidl function y cos. Trnsltion nd diltion trnsformtions lso pply to circulr function sinusoids. 1. For y 3 4 cos ( 6), find. The horizontl diltion. The verticl diltion c. The horizontl trnsltion d. The verticl trnsltion 13. For sinusoids, list the specil nmes given to. The horizontl diltion. The verticl diltion c. The horizontl trnsltion d. The verticl trnsltion v u 488 Chpter 9: Tringle Trigonometry

51 To use sinusoids s mthemticl models, you lerned to write prticulr eqution from the grph. y Figure Write the prticulr eqution for the sinusoid in Figure If the grph in Prolem 14 were plotted on wide-enough domin, predict y for For the sinusoid in Prolem 14, find lgericlly the first three positive vlues of if y Show grphiclly tht the three vlues you found in Prolem 16 re correct. 18. Stellite Prolem : Assume tht in Prolem 1, the stellite s distnce vries sinusoidlly with time. Suppose tht the stellite is closest, 1000 mi from you, t time t 0 min. Hlf period lter, t t 0 min, it is t its mimum distnce from you, 9000 mi. Write prticulr eqution for distnce, in thousnds of miles, s function of time. Rdins gve you convenient wy to nlyze the motion of two or more rotting ojects. 19. Figure 9-9 shows -cm-rdius ger on mchine tool driving 1-cm-rdius ger. The design engineers wnt the smller ger s teeth to hve liner velocity 10 cm/s. cm Figure cm 3. sin u 13, cos u 1 13, tn u 1, cot u 1, sec u 13 1, csc u y sin Sinusoidl 8. p; p ; p ; p 4 0. Wht will e the ngulr velocity of the smller ger in rdins per second? In revolutions per minute?. Wht will e the liner velocity of the lrger ger s teeth? c. At how mny revolutions per minute will the lrger ger rotte? 0. Stellite Prolem 3: Figure 9-9c shows the stellite of Prolems 1 nd 18 in n orit with rdius 000 mi round Erth. Erth is ssumed to hve rdius 4000 mi. As in Prolem 18, ssume tht it tkes 100 min for the stellite to mke one complete orit round Erth.. Wht is the stellite s ngulr velocity in rdins per minute?. How fst is it going in miles per hour? c. Wht interesting connection do you notice etween the ngulr velocity in prt nd the sinusoidl eqution in Prolem 18? Stellite 000 mi y You 4000 mi Figure 9-9c Net For you cos lerned 0 some ( properties p 1 pn) of trigonometric nd circulr See pge functions. 101 for nswers to 1. There Prolems re three nd kinds 10 of properties tht involve just one rgument. Write the nme of ech kind of property, nd give n emple of ech.. Use the properties in Prolem 1 to prove tht this eqution is n identity. Wht restrictions re there on the domin of? s e c sin 1 tn 4 sin c o s 4 Section 9-9: y cos c. 6 1d p or 360 times horizontl diltion is the period. 13. Amplitude PROBLEM NOTES 13c. Phse displcement or phse shift 13d. Sinusoidl is 14. y 1 3 cos p ( 1) 1. y , 9.7, y 18. d 4 cos 0 p rd/s; rev/min cm/s 19c rev/min 0. p rd/min rd/min p mi/h 18,80 mi/h 10 0c. The ngulr velocity is the coefficient, B, of the rgument. 1. Reciprocl properties: sec u 1 cos u, csc u 1 sin u, cot u 1 tn u Quotient properties: tn u sin u cos u, cot u cos u sin u Pythgoren properties: si n u 1 cos u 1, tn u 1 1 sec u, 1 1 cot u csc u. s e c sin 1 tn 4 sin cos 1 sin 4 cos 4 sin cos 1 sin 4 cos 4 cos 4 sin cos 1 sin 4 cos 4 sin (cos 1 sin ) sin cos 4 cos 4 For cos 0 ( p 1 pn) 1 See pge 101 for nswers to Prolems nd 10. Section 9-9: Cumultive Review, Chpters

52 Prolem Notes (continued) 3. cos( y) cos cos y 1 sin sin y Cosine of first, cosine of second, plus sine of first, sine of second 4. cos sin 6. cos(90 u) cos 90 cos u 1 sin 90 sin u 0? cos u 1 1? sin u sin u; cos(34 ) cos(90 6 ) sin 6 6. A ; D [ 3 cos u 1 4 sin u cos(u ) 7. 6 sin u 6 sin(u 1 u) 6(sin u cos u 1 cos u sin u) 6? sin u cos u 1 sin u cos u 8. cos 1 sin, so sin 1 1 cos, which is sinusoid. 9. Lrger sinusoid: y 3 cos 6u; Smller sinusoid: y sin 30u; Comined: y 3 cos 6u 1 sin 30u 30. Lrger sinusoid: y sin ; Smller sinusoid: y cos 1; Comined: y sin cos y cos 1u 1 cos 19u 3. The period of cos 0u is 18 ; the period of cos u is 360. These re very different. In the nswer, the period of cos 1u is ; the period of cos 19u is These re nerly equl. 33. u y pn rd or pn rd 3. See Figure 9-9f in the student tet. Possile prmetric equtions: cos t, y t 36. Domin is 1 1; Rnge is 0 y 37. u 63.4, 43.4, 43.4, Other properties involve functions of composite rgument. Write the composite rgument property for cos( y). Then epress this property verlly. 4. Show numericlly tht cos 34 sin 6.. Use the property in Prolem 3 to prove tht the eqution cos(90 ) sin is n identity. How does this eplin the result in Prolem 4? The properties cn e used to eplin why certin comintions of grphs come out the wy they do. 6. Show tht the function y 3 cos 4 sin is sinusoid y finding lgericlly the mplitude nd phse displcement with respect to y cos nd writing y s single sinusoid. 7. The function y 1 sin cos is equivlent to the sinusoid y 6 sin. Prove lgericlly tht this is true y pplying the composite rgument property to sin. 8. Write the doule rgument property epressing cos in terms of sin lone. Use this property to show lgericlly tht the grph of y sin is sinusoid. Sums nd products of sinusoids with different periods hve interesting wve ptterns. By using hrmonic nlysis, you cn write equtions of the two sinusoids tht were dded or multiplied. 9. Find the prticulr eqution for the function in Figure 9-9d. y 490 Chpter 9: Figure 9-9d 30. Find the prticulr eqution for the function in Figure 9-9e. y Figure 9-9e A product of sinusoids with very different periods cn e trnsformed to sum of sinusoids with nerly equl periods. 31. Trnsform the function y cos 0 cos into sum of two cosine functions. 3. Find the periods of the two sinusoids in the eqution given in Prolem 31 nd the periods of the two sinusoids in the nswer. Wht cn you tell out reltive sizes of the periods of the two sinusoids in the given eqution nd out reltive sizes of the periods of the sinusoids in the nswer? Trigonometric nd circulr functions re periodic, so there re mny vlues of or tht give the sme vlue of y. Thus, the inverses of these functions re not functions. 33. Find the (one) vlue of the inverse trigonometric function tn Find the generl solution of the inverse trigonometric reltion rcsin Chpter 9: Tringle Trigonometry

53 Prmetric functions mke it possile to plot the grphs of inverse circulr reltions. 6 y 1 Figure 9-9f 3. Use prmetric functions to crete the grph of y rccos, s shown in Figure 9-9f. 36. The inverse trigonometric function y cos 1 is the principl rnch of y rccos. Define the domin nd rnge of y cos Find the first four positive vlues of if rctn. Lst, you studied tringle nd vector prolems. 38. Stte the lw of cosines. 39. Stte the lw of sines. 40. Stte the re formul for tringle given two sides nd the included ngle. 41. If tringle hs sides 6 ft, 7 ft, nd 1 ft, find the mesure of the lrgest ngle. 4. Find the re of the tringle in Prolem 41 using Hero s formul. 43. Given 3 i 4 j nd i 1 j,. Find the resultnt vector,, in terms of its components.. Find the mgnitude nd ngle in stndrd position of the resultnt vector. c. Sketch figure to show dded geometriclly, hed-to-til. d. Is this true or flse? Eplin why your nswer is resonle. The tringle properties cn e used to show tht periodic functions tht look like sinusoids my not ctully e sinusoids. 44. Stellite Prolem 4: In Prolem 18, you ssumed tht the distnce etween you nd the stellite ws sinusoidl function of time. In this prolem you will get more ccurte mthemticl model.. Use the lw of cosines nd the distnces in Figure 9-9g to find y s function of ngle, in rdins. Stellite 000 mi y Figure 9-9g 4000 mi You. Use the fct tht it tkes 100 min for the stellite to mke one orit to write the eqution for y s function of time t. Assume tht 0 t time t 0 min. c. Plot the eqution from prt nd the eqution from Prolem 18 on the sme screen, thus showing tht the functions hve the sme high points, low points, nd period ut tht the eqution from prt is not sinusoid. Section 9-9: In nabc, c 1 cos C (nd similrly for nd ). The squre of one side of tringle is the sum of the squres of the other two sides minus twice their product times the cosine of the ngle etween them. 39. In nabc, sin A sin B c sin C. The length of one side of tringle is to the sine of the ngle opposite it s the length of ny other side is to the sine of the ngle opposite tht side. 40. A nabc 1 sin C 1 c sin A 1 c sin B. The re of tringle is 1 the product of ny two sides nd the sine of the ngle etween them ft 43. i 1 16 j 43. r 16.1; u c. v 4 43d. Flse. This is true only if nd re t the sme ngle. 44. In units of 1000 miles: 44. y cos 44. y cos p t. 44c. The dshed curve represents the eqution from Prolem 18. y (1000 mi) 1 10 u t (min) Section 9-9: Cumultive Review, Chpters 9 491