Right Triangles and Trigonometry

Transcription

1 9 Right Tringles nd Trigonometry 9.1 The Pythgoren Theorem 9. Specil Right Tringles 9.3 Similr Right Tringles 9.4 The Tngent Rtio 9.5 The Sine nd osine Rtios 9.6 Solving Right Tringles 9.7 Lw of Sines nd Lw of osines Lening Tower of Pis (p. 514) Skiing (p. 497) SEE the ig Ide Wshington Monument (p. 491) Rock Wll (p. 481) Fire Escpe (p. 469)

2 Mintining Mthemticl Proficiency Using Properties of Rdicls Emple 1 Simplify = 64 Emple Simplify 4. 5 Fctor using the gretest perfect squre fctor. = 64 Product Property of Squre Roots = 8 Simplify. 4 = = = Multiply y 5. 5 Product Property of Squre Roots Simplify. Simplify the epression Solving Proportions Emple 3 Solve 10 = = 3 Write the proportion. = 10 3 ross Products Property = 30 Multiply. = 30 Divide ech side y. = 15 Simplify. Solve the proportion = = = = = = STRT RESONING The Product Property of Squre Roots llows you to simplify the squre root of product. re you le to simplify the squre root of sum? of difference? Eplin. Dynmic Solutions ville t igidesmth.com 461

3 Mthemticl Prctices ttending to Precision ore oncept Stndrd Position for Right Tringle In unit circle trigonometry, right tringle is in stndrd position when: 1. The hypotenuse is rdius of the circle of rdius 1 with center t the origin.. One leg of the right tringle lies on the -is. 3. The other leg of the right tringle is perpendiculr to the -is. Mthemticlly profi cient students epress numericl nswers precisely y Drwing n Isosceles Right Tringle in Stndrd Position Use dynmic geometry softwre to construct n isosceles right tringle in stndrd position. Wht re the ect coordintes of its vertices? Smple Points (0, 0) (0.71, 0.71) (0.71, 0) Segments = 1 = 0.71 = 0.71 ngle m = 45 To determine the ect coordintes of the vertices, lel the length of ech leg. y the Pythgoren Theorem, which you will study in Section 9.1, + = 1. Solving this eqution yields = 1, or. So, the ect coordintes of the vertices re (0, 0), (, ), nd ( ), 0. Monitoring Progress 1. Use dynmic geometry softwre to construct right tringle with cute ngle mesures of 30 nd 60 in stndrd position. Wht re the ect coordintes of its vertices?. Use dynmic geometry softwre to construct right tringle with cute ngle mesures of 0 nd 70 in stndrd position. Wht re the pproimte coordintes of its vertices? 46 hpter 9 Right Tringles nd Trigonometry

4 9.1 The Pythgoren Theorem Essentil Question How cn you prove the Pythgoren Theorem? Proving the Pythgoren Theorem without Words Work with prtner.. Drw nd cut out right tringle with legs nd, nd hypotenuse c.. Mke three copies of your right tringle. rrnge ll four tringles to form lrge squre, s shown. c. Find the re of the lrge squre in terms of,, nd c y summing the res of the tringles nd the smll squre. c c c c d. opy the lrge squre. Divide it into two smller squres nd two eqully-sized rectngles, s shown. e. Find the re of the lrge squre in terms of nd y summing the res of the rectngles nd the smller squres. f. ompre your nswers to prts (c) nd (e). Eplin how this proves the Pythgoren Theorem. Proving the Pythgoren Theorem Work with prtner.. Drw right tringle with legs nd, nd hypotenuse c, s shown. Drw the ltitude from to. Lel the lengths, s shown. RESONING STRTLY To e proficient in mth, you need to know nd fleily use different properties of opertions nd ojects. h c d c D. Eplin why, D, nd D re similr. d c. Write two-column proof using the similr tringles in prt () to prove tht + = c. ommunicte Your nswer 3. How cn you prove the Pythgoren Theorem? 4. Use the Internet or some other resource to find wy to prove the Pythgoren Theorem tht is different from Eplortions 1 nd. Section 9.1 The Pythgoren Theorem 463

5 9.1 Lesson Wht You Will Lern ore Voculry Pythgoren triple, p. 464 Previous right tringle legs of right tringle hypotenuse Use the Pythgoren Theorem. Use the onverse of the Pythgoren Theorem. lssify tringles. Using the Pythgoren Theorem One of the most fmous theorems in mthemtics is the Pythgoren Theorem, nmed for the ncient Greek mthemticin Pythgors. This theorem descries the reltionship etween the side lengths of right tringle. Theorem Theorem 9.1 Pythgoren Theorem In right tringle, the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs. Proof Eplortions 1 nd, p. 463; E. 39, p. 484 c c = + STUDY TIP You my find it helpful to memorize the sic Pythgoren triples, shown in old, for stndrdized tests. Pythgoren triple is set of three positive integers,, nd c tht stisfy the eqution c = +. ore oncept ommon Pythgoren Triples nd Some of Their Multiples 3, 4, 5 6, 8, 10 9, 1, 15 3, 4, 5 5, 1, 13 10, 4, 6 15, 36, 39 5, 1, 13 8, 15, 17 16, 30, 34 4, 45, 51 8, 15, 17 7, 4, 5 14, 48, 50 1, 7, 75 7, 4, 5 The most common Pythgoren triples re in old. The other triples re the result of multiplying ech integer in old-fced triple y the sme fctor. Using the Pythgoren Theorem Find the vlue of. Then tell whether the side lengths form Pythgoren triple. c = + = = = 169 = 13 Pythgoren Theorem Sustitute. Multiply. dd. Find the positive squre root. 5 1 The vlue of is 13. ecuse the side lengths 5, 1, nd 13 re integers tht stisfy the eqution c = +, they form Pythgoren triple. 464 hpter 9 Right Tringles nd Trigonometry

6 Using the Pythgoren Theorem Find the vlue of. Then tell whether the side lengths form Pythgoren triple. 7 c = + Pythgoren Theorem 14 = 7 + Sustitute. 196 = 49 + Multiply. 147 = Sutrct 49 from ech side. 147 = Find the positive squre root = Product Property of Squre Roots 7 3 = Simplify. 14 The vlue of is 7 3. ecuse 7 3 is not n integer, the side lengths do not form Pythgoren triple. Solving Rel-Life Prolem The skyscrpers shown re connected y skywlk with support ems. Use the Pythgoren Theorem to pproimte the length of ech support em. Ech support em forms the hypotenuse of right tringle. The right tringles re congruent, so the support ems re the sme length. = (3.6) + (47.57) Pythgoren Theorem = (3.6) + (47.57) Find the positive squre root m Use clcultor to pproimte. The length of ech support em is out 5.95 meters. 3.6 m support ems m Monitoring Progress Help in English nd Spnish t igidesmth.com Find the vlue of. Then tell whether the side lengths form Pythgoren triple n nemometer is device used to mesure wind speed. The nemometer shown is ttched to the top of pole. Support wires re ttched to the pole 5 feet ove the ground. Ech support wire is 6 feet long. How fr from the se of the pole is ech wire ttched to the ground? 6 ft d 5 ft Section 9.1 The Pythgoren Theorem 465

7 Using the onverse of the Pythgoren Theorem The converse of the Pythgoren Theorem is lso true. You cn use it to determine whether tringle with given side lengths is right tringle. Theorem Theorem 9. onverse of the Pythgoren Theorem If the squre of the length of the longest side of tringle is equl to the sum of the squres of the lengths of the other two sides, then the tringle is right tringle. If c = +, then is right tringle. Proof E. 39, p. 470 c Verifying Right Tringles Tell whether ech tringle is right tringle. USING TOOLS STRTEGILLY Use clcultor to determine tht is the length of the longest side in prt () Let c represent the length of the longest side of the tringle. heck to see whether the side lengths stisfy the eqution c = +.. ( 113 ) =? =? = 113 The tringle is right tringle.. ( 4 95 ) =? ( 95 ) =? =? The tringle is not right tringle. Monitoring Progress Help in English nd Spnish t igidesmth.com Tell whether the tringle is right tringle hpter 9 Right Tringles nd Trigonometry

8 lssifying Tringles The onverse of the Pythgoren Theorem is used to determine whether tringle is right tringle. You cn use the theorem elow to determine whether tringle is cute or otuse. Theorem Theorem 9.3 Pythgoren Inequlities Theorem For ny, where c is the length of the longest side, the following sttements re true. If c < +, then is cute. If c > +, then is otuse. c c REMEMER The Tringle Inequlity Theorem (Theorem 6.11) on pge 339 sttes tht the sum of the lengths of ny two sides of tringle is greter thn the length of the third side. lssifying Tringles Verify tht segments with lengths of 4.3 feet, 5. feet, nd 6.1 feet form tringle. Is the tringle cute, right, or otuse? c < + Proof Es. 4 nd 43, p. 470 Step 1 Use the Tringle Inequlity Theorem (Theorem 6.11) to verify tht the segments form tringle. c > >? >? >? > > > 4.3 The segments with lengths of 4.3 feet, 5. feet, nd 6.1 feet form tringle. Step lssify the tringle y compring the squre of the length of the longest side with the sum of the squres of the lengths of the other two sides. c + ompre c with Sustitute Simplify < c is less thn +. The segments with lengths of 4.3 feet, 5. feet, nd 6.1 feet form n cute tringle. Monitoring Progress Help in English nd Spnish t igidesmth.com 6. Verify tht segments with lengths of 3, 4, nd 6 form tringle. Is the tringle cute, right, or otuse? 7. Verify tht segments with lengths of.1,.8, nd 3.5 form tringle. Is the tringle cute, right, or otuse? Section 9.1 The Pythgoren Theorem 467

9 9.1 Eercises Dynmic Solutions ville t igidesmth.com Voculry nd ore oncept heck 1. VOULRY Wht is Pythgoren triple?. DIFFERENT WORDS, SME QUESTION Which is different? Find oth nswers. Find the length of the longest side. Find the length of the hypotenuse. 3 Find the length of the longest leg. 4 Find the length of the side opposite the right ngle. Monitoring Progress nd Modeling with Mthemtics In Eercises 3 6, find the vlue of. Then tell whether the side lengths form Pythgoren triple. (See Emple 1.) In Eercises 7 10, find the vlue of. Then tell whether the side lengths form Pythgoren triple. (See Emple.) ERROR NLYSIS In Eercises 11 nd 1, descrie nd correct the error in using the Pythgoren Theorem (Theorem 9.1) c = + = = (7 + 4) = 31 = 31 c = + = = = 776 = hpter 9 Right Tringles nd Trigonometry

10 13. MODELING WITH MTHEMTIS The fire escpe forms right tringle, s shown. Use the Pythgoren Theorem (Theorem 9.1) to pproimte the distnce etween the two pltforms. (See Emple 3.) In Eercises 1 8, verify tht the segment lengths form tringle. Is the tringle cute, right, or otuse? (See Emple 5.) 1. 10, 11, nd 14. 6, 8, nd ft 3. 1, 16, nd , 0, nd , 6.7, nd , 8., nd , 30, nd , 15, nd ft 14. MODELING WITH MTHEMTIS The ckord of the sketll hoop forms right tringle with the supporting rods, s shown. Use the Pythgoren Theorem (Theorem 9.1) to pproimte the distnce etween the rods where they meet the ckord. 9. MODELING WITH MTHEMTIS In sell, the lengths of the pths etween consecutive ses re 90 feet, nd the pths form right ngles. The plyer on first se tries to stel second se. How fr does the ll need to trvel from home plte to second se to get the plyer out? 30. RESONING You re mking cnvs frme for pinting using stretcher rs. The rectngulr pinting will e 10 inches long nd 8 inches wide. Using ruler, how cn you e certin tht the corners of the frme re 90? 13.4 in. 9.8 in. In Eercises 15 0, tell whether the tringle is right tringle. (See Emple 4.) In Eercises 31 34, find the re of the isosceles tringle m 10 cm h 16 m h 17 m 10 cm ft h 3 ft 0 ft cm 50 m h 50 m m Section 9.1 The Pythgoren Theorem 469

11 35. NLYZING RELTIONSHIPS Justify the Distnce Formul using the Pythgoren Theorem (Thm. 9.1). 36. HOW DO YOU SEE IT? How do you know is right ngle? PROLEM SOLVING You re mking kite nd need to figure out how much inding to uy. You need the inding for the perimeter of the kite. The inding comes in 1 in. pckges of two yrds. How mny pckges should you uy? 38. PROVING THEOREM Use the Pythgoren Theorem (Theorem 9.1) to prove the Hypotenuse-Leg (HL) ongruence Theorem (Theorem 5.9). 39. PROVING THEOREM Prove the onverse of the Pythgoren Theorem (Theorem 9.). (Hint: Drw with side lengths,, nd c, where c is the length of the longest side. Then drw right tringle with side lengths,, nd, where is the length of the hypotenuse. ompre lengths c nd.) 40. THOUGHT PROVOKING onsider two positive integers m nd n, where m > n. Do the following epressions produce Pythgoren triple? If yes, prove your nswer. If no, give counteremple. 8 mn, m n, m + n 15 in. 1 in. 0 in. 41. MKING N RGUMENT Your friend clims 7 nd 75 cnnot e prt of Pythgoren triple ecuse does not equl positive integer squred. Is your friend correct? Eplin your resoning. 4. PROVING THEOREM opy nd complete the proof of the Pythgoren Inequlities Theorem (Theorem 9.3) when c < +. Given In, c < +, where c is the length of the longest side. PQR hs side lengths,, nd, where is the length of the hypotenuse, nd R is right ngle. Prove is n cute tringle. STTEMENTS c Q R 1. In, c < +, where c is the length of the longest side. PQR hs side lengths,, nd, where is the length of the hypotenuse, nd R is right ngle. P RESONS =. 3. c < c < 4. Tke the positive squre root of ech side. 5. m R = m < m R 6. onverse of the Hinge Theorem (Theorem 6.13) 7. m < is n cute ngle is n cute tringle PROVING THEOREM Prove the Pythgoren Inequlities Theorem (Theorem 9.3) when c > +. (Hint: Look ck t Eercise 4.) Mintining Mthemticl Proficiency Simplify the epression y rtionlizing the denomintor. (Skills Review Hndook) Reviewing wht you lerned in previous grdes nd lessons hpter 9 Right Tringles nd Trigonometry

12 9. Specil Right Tringles Essentil Question Wht is the reltionship mong the side lengths of tringles? tringles? Side Rtios of n Isosceles Right Tringle TTENDING TO PREISION To e proficient in mth, you need to epress numericl nswers with degree of precision pproprite for the prolem contet. Work with prtner.. Use dynmic geometry softwre to construct n isosceles right tringle with leg length of 4 units.. Find the cute ngle mesures. Eplin why this tringle is clled tringle. c. Find the ect rtios of the side lengths (using squre roots). = = = d. Repet prts () nd (c) for severl other isosceles right tringles. Use your results to write conjecture out the rtios of the side lengths of n isosceles right tringle. Smple Points (0, 4) (4, 0) (0, 0) Segments = 5.66 = 4 = 4 ngles m = 45 m = 45 Side Rtios of Tringle Work with prtner.. Use dynmic geometry softwre to construct right tringle with cute ngle mesures of 30 nd 60 ( tringle), where the shorter leg length is 3 units.. Find the ect rtios Smple 5 of the side lengths (using squre roots). = = = c. Repet prts () nd () for severl other tringles. Use your results to write conjecture out the rtios of the side lengths of tringle. ommunicte Your nswer 3. Wht is the reltionship mong the side lengths of tringles? tringles? Points (0, 5.0) (3, 0) (0, 0) Segments = 6 = 3 = 5.0 ngles m = 30 m = 60 Section 9. Specil Right Tringles 471

13 9. Lesson Wht You Will Lern ore Voculry Previous isosceles tringle Find side lengths in specil right tringles. Solve rel-life prolems involving specil right tringles. Finding Side Lengths in Specil Right Tringles tringle is n isosceles right tringle tht cn e formed y cutting squre in hlf digonlly. REMEMER n epression involving rdicl with inde is in simplest form when no rdicnds hve perfect squres s fctors other thn 1, no rdicnds contin frctions, nd no rdicls pper in the denomintor of frction. Theorem Theorem Tringle Theorem In tringle, the hypotenuse is times s long s ech leg. Finding Side Lengths in Tringles Find the vlue of. Write your nswer in simplest form Proof E. 19, p. 476 hypotenuse = leg. y the Tringle Sum Theorem (Theorem 5.1), the mesure of the third ngle must e 45, so the tringle is tringle. hypotenuse = leg Tringle Theorem = 8 = 8 The vlue of is 8. Sustitute. Simplify.. y the se ngles Theorem (Theorem 5.6) nd the orollry to the Tringle Sum Theorem (orollry 5.1), the tringle is tringle. hypotenuse = leg Tringle Theorem 5 = Sustitute. 5 = Divide ech side y. 5 = Simplify. The vlue of is hpter 9 Right Tringles nd Trigonometry

14 Theorem Theorem Tringle Theorem In tringle, the hypotenuse is twice s long s the shorter leg, nd the longer leg is 3 times s long s the shorter leg. Proof E. 1, p hypotenuse = shorter leg longer leg = shorter leg 3 REMEMER ecuse the ngle opposite 9 is lrger thn the ngle opposite, the leg with length 9 is longer thn the leg with length y the Tringle Lrger ngle Theorem (Theorem 6.10). Finding Side Lengths in Tringle Find the vlues of nd y. Write your nswer in simplest form. Step 1 Find the vlue of. longer leg = shorter leg = 3 Sustitute Tringle Theorem 9 = Divide ech side y = Multiply y = Multiply frctions = Simplify. y The vlue of is 3 3. Step Find the vlue of y. hypotenuse = shorter leg y = 3 3 y = Tringle Theorem Sustitute. Simplify. The vlue of y is 6 3. Monitoring Progress Help in English nd Spnish t igidesmth.com Find the vlue of the vrile. Write your nswer in simplest form. 1.. y h 4 Section 9. Specil Right Tringles 473

15 Solving Rel-Life Prolems Modeling with Mthemtics The rod sign is shped like n equilterl tringle. Estimte the re of the sign y finding the re of the equilterl tringle. First find the height h of the tringle y dividing it into two tringles. The length of the longer leg of one of these tringles is h. The length of the shorter leg is 18 inches. h = 18 3 = Tringle Theorem Use h = 18 3 to find the re of the equilterl tringle. re = 1 h = 1 (36) ( 18 3 ) The re of the sign is out 561 squre inches. 36 in. YIELD 18 in. 18 in h 36 in. 36 in. Finding the Height of Rmp tipping pltform is rmp used to unlod trucks. How high is the end of n 80-foot rmp when the tipping ngle is 30? 45? rmp 80 ft height of rmp tipping ngle 14 ft 60 When the tipping ngle is 30, the height h of the rmp is the length of the shorter leg of tringle. The length of the hypotenuse is 80 feet. 80 = h Tringle Theorem 40 = h Divide ech side y. When the tipping ngle is 45, the height h of the rmp is the length of leg of tringle. The length of the hypotenuse is 80 feet. 80 = h Tringle Theorem 80 = h Divide ech side y h Use clcultor. When the tipping ngle is 30, the rmp height is 40 feet. When the tipping ngle is 45, the rmp height is out 56 feet 7 inches. Monitoring Progress Help in English nd Spnish t igidesmth.com 5. The logo on recycling in resemles n equilterl tringle with side lengths of 6 centimeters. pproimte the re of the logo. 6. The ody of dump truck is rised to empty lod of snd. How high is the 14-foot-long ody from the frme when it is tipped upwrd y 60 ngle? 474 hpter 9 Right Tringles nd Trigonometry

16 9. Eercises Dynmic Solutions ville t igidesmth.com Voculry nd ore oncept heck 1. VOULRY Nme two specil right tringles y their ngle mesures.. WRITING Eplin why the cute ngles in n isosceles right tringle lwys mesure 45. Monitoring Progress nd Modeling with Mthemtics In Eercises 3 6, find the vlue of. Write your nswer in simplest form. (See Emple 1.) In Eercises 7 10, find the vlues of nd y. Write your nswers in simplest form. (See Emple.) y y ERROR NLYSIS In Eercises 11 nd 1, descrie nd correct the error in finding the length of the hypotenuse y the Tringle Sum Theorem (Theorem 5.1), the mesure of the third ngle must e 60. So, the tringle is tringle. hypotenuse = shorter leg 3 = 7 3 y y y the Tringle Sum Theorem (Theorem 5.1), the mesure of the third ngle must e 45. So, the tringle is tringle. hypotenuse = leg leg = 5 So, the length of the hypotenuse is 5 units. In Eercises 13 nd 14, sketch the figure tht is descried. Find the indicted length. Round deciml nswers to the nerest tenth. 13. The side length of n equilterl tringle is 5 centimeters. Find the length of n ltitude. 14. The perimeter of squre is 36 inches. Find the length of digonl. In Eercises 15 nd 16, find the re of the figure. Round deciml nswers to the nerest tenth. (See Emple 3.) ft m 60 5 m 17. PROLEM SOLVING Ech hlf of the drwridge is out 84 feet long. How high does the drwridge rise when is 30? 45? 60? (See Emple 4.) 84 ft 5 m 4 m So, the length of the hypotenuse is 7 3 units. Section 9. Specil Right Tringles 475

17 18. MODELING WITH MTHEMTIS nut is shped like regulr hegon with side lengths of 1 centimeter. Find the vlue of. (Hint: regulr hegon cn e divided into si congruent tringles.) 1 cm. THOUGHT PROVOKING The digrm elow is clled the illes rectngle. Ech tringle in the digrm hs rtionl ngle mesures nd ech side length contins t most one squre root. Lel the sides nd ngles in the digrm. Descrie the tringles. 19. PROVING THEOREM Write prgrph proof of the Tringle Theorem (Theorem 9.4). Given DEF is D tringle. Prove The hypotenuse is 45 times s long 45 s ech leg. F E 0. HOW DO YOU SEE IT? The digrm shows prt of the Wheel of Theodorus WRITING Descrie two wys to show tht ll isosceles right tringles re similr to ech other. 4. MKING N RGUMENT Ech tringle in the digrm is tringle. t Stge 0, the legs of the tringle re ech 1 unit long. Your rother clims the lengths of the legs of the tringles dded re hlved t ech stge. So, the length of leg of tringle dded in Stge 8 will e 1 unit. Is your 56 rother correct? Eplin your resoning Stge 0 Stge 1 Stge. Which tringles, if ny, re tringles?. Which tringles, if ny, re tringles? 1. PROVING THEOREM Write prgrph proof of the Tringle Theorem (Theorem 9.5). (Hint: onstruct JML congruent to JKL.) Given JKL is tringle. K Prove The hypotenuse is twice s long s the shorter 30 J leg, nd the longer leg is 3 times s long s the shorter leg. 60 L M Stge 3 Stge 4 5. USING STRUTURE TUV is tringle, where two vertices re U(3, 1) nd V( 3, 1), UV is the hypotenuse, nd point T is in Qudrnt I. Find the coordintes of T. Mintining Mthemticl Proficiency Find the vlue of. (Section 8.1) 6. DEF LMN 7. QRS Reviewing wht you lerned in previous grdes nd lessons E 1 D N 0 F 30 L M S 4 R Q 476 hpter 9 Right Tringles nd Trigonometry

18 9.3 Similr Right Tringles Essentil Question How re ltitudes nd geometric mens of right tringles relted? Writing onjecture Work with prtner.. Use dynmic geometry softwre to construct right, s shown. Drw D so tht it is n ltitude from the right ngle to the hypotenuse of D Points (0, 5) (8, 0) (0, 0) D(.5, 3.6) Segments = 9.43 = 8 = 5 ONSTRUTING VILE RGUMENTS To e proficient in mth, you need to understnd nd use stted ssumptions, definitions, nd previously estlished results in constructing rguments.. The geometric men of two positive numers nd is the positive numer tht stisfies =. is the geometric men of nd. Write proportion involving the side lengths of D nd D so tht D is the geometric men of two of the other side lengths. Use similr tringles to justify your steps. c. Use the proportion you wrote in prt () to find D. d. Generlize the proportion you wrote in prt (). Then write conjecture out how the geometric men is relted to the ltitude from the right ngle to the hypotenuse of right tringle. Work with prtner. Use spredsheet to find the rithmetic men nd the geometric men of severl pirs of positive numers. ompre the two mens. Wht do you notice? ommunicte Your nswer ompring Geometric nd rithmetic Mens D rithmetic Men Geometric Men How re ltitudes nd geometric mens of right tringles relted? Section 9.3 Similr Right Tringles 477

19 9.3 Lesson Wht You Will Lern ore Voculry geometric men, p. 480 Previous ltitude of tringle similr figures Identify similr tringles. Solve rel-life prolems involving similr tringles. Use geometric mens. Identifying Similr Tringles When the ltitude is drwn to the hypotenuse of right tringle, the two smller tringles re similr to the originl tringle nd to ech other. Theorem Theorem 9.6 Right Tringle Similrity Theorem If the ltitude is drwn to the hypotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd to ech other. D D, D, nd D D. Proof E. 45, p. 484 D D Identifying Similr Tringles Identify the similr tringles in the digrm. U S R T Sketch the three similr right tringles so tht the corresponding ngles nd sides hve the sme orienttion. T S S T U R U R T TSU RTU RST Monitoring Progress Help in English nd Spnish t igidesmth.com Identify the similr tringles. 1. Q. E H F T S R G 478 hpter 9 Right Tringles nd Trigonometry

20 Solving Rel-Life Prolems Modeling with Mthemtics roof hs cross section tht is right tringle. The digrm shows the pproimte dimensions of this cross section. Find the height h of the roof. Y 5.5 m h 3.1 m Z 6.3 m W X 1. Understnd the Prolem You re given the side lengths of right tringle. You need to find the height of the roof, which is the ltitude drwn to the hypotenuse.. Mke Pln Identify ny similr tringles. Then use the similr tringles to write proportion involving the height nd solve for h. 3. Solve the Prolem Identify the similr tringles nd sketch them. Z Z OMMON ERROR Notice tht if you tried to write proportion using XYW nd YZW, then there would e two unknowns, so you would not e le to solve for h. 3.1 m X Y h W 5.5 m Y h W X 6.3 m XYW YZW XZY ecuse XYW XZY, you cn write proportion. 3.1 m Y 5.5 m YW ZY = XY XZ orresponding side lengths of similr tringles re proportionl. h 5.5 = Sustitute. h.7 Multiply ech side y 5.5. The height of the roof is out.7 meters. 4. Look ck ecuse the height of the roof is leg of right YZW nd right XYW, it should e shorter thn ech of their hypotenuses. The lengths of the two hypotenuses re YZ = 5.5 nd XY = 3.1. ecuse.7 < 3.1, the nswer seems resonle. Monitoring Progress Find the vlue of. Help in English nd Spnish t igidesmth.com 3. E 3 G H 4 5 F 4. J 13 1 K L 5 M Section 9.3 Similr Right Tringles 479

21 Using Geometric Men ore oncept Geometric Men The geometric men of two positive numers nd is the positive numer tht stisfies =. So, = nd =. Finding Geometric Men Find the geometric men of 4 nd 48. = Definition of geometric men = 4 48 Sustitute 4 for nd 48 for. = 4 48 Tke the positive squre root of ech side. = 4 4 Fctor. = 4 Simplify. The geometric men of 4 nd 48 is D In right, ltitude D is drwn to the hypotenuse, forming two smller right tringles tht re similr to. From the Right Tringle Similrity Theorem, you know tht D D. ecuse the tringles re similr, you cn write nd simplify the following proportions involving geometric mens. D D = D D D = D = D = D D = D = D D D Theorems Theorem 9.7 Geometric Men (ltitude) Theorem In right tringle, the ltitude from the right ngle to the hypotenuse divides the hypotenuse into two segments. The length of the ltitude is the geometric men of the lengths of the two segments of the hypotenuse. D Proof E. 41, p. 484 D = D D Theorem 9.8 Geometric Men (Leg) Theorem In right tringle, the ltitude from the right ngle to the hypotenuse divides the hypotenuse into two segments. The length of ech leg of the right tringle is the geometric men of the lengths of the hypotenuse nd the segment of the hypotenuse tht is djcent to the leg. D = D = D Proof E. 4, p hpter 9 Right Tringles nd Trigonometry

22 Using Geometric Men OMMON ERROR In Emple 4(), the Geometric Men (Leg) Theorem gives y = (5 + ), not y = 5 (5 + ), ecuse the side with length y is djcent to the segment with length. Find the vlue of ech vrile y 5. pply the Geometric Men. pply the Geometric Men (ltitude) Theorem. (Leg) Theorem. = 6 3 y = (5 + ) = 18 y = 7 = 18 y = 14 = 9 y = 14 = 3 The vlue of y is 14. The vlue of is 3. Using Indirect Mesurement To find the cost of instlling rock wll in your school gymnsium, you need to find the height of the gym wll. You use crdord squre to line up the top nd ottom om of the gym wll. Your friend mesures the verticl distnce from the ground to your eye nd the horizontl distnce from you to the gym wll. pproimte the height of the gym wll. y the Geometric Men (ltitude) Theorem, you know tht 8.5 is the geometric men of w nd = w 5 Geometric Men (ltitude) Theorem 7.5 = 5w Squre = w Divide ech side y 5. The height of the wll is 5 + w = = feet. 8.5 ft w ft 5 ft 4 9 Monitoring Progress Find the geometric men of the two numers. Help in English nd Spnish t igidesmth.com 5. 1 nd nd nd Find the vlue of in the tringle t the left. 9. WHT IF? In Emple 5, the verticl distnce from the ground to your eye is 5.5 feet nd the distnce from you to the gym wll is 9 feet. pproimte the height of the gym wll. Section 9.3 Similr Right Tringles 481

23 9.3 Eercises Dynmic Solutions ville t igidesmth.com Voculry nd ore oncept heck 1. OMPLETE THE SENTENE If the ltitude is drwn to the hypotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd.. WRITING In your own words, eplin geometric men. Monitoring Progress nd Modeling with Mthemtics In Eercises 3 nd 4, identify the similr tringles. (See Emple 1.) 3. F E In Eercises 11 18, find the geometric men of the two numers. (See Emple 3.) nd nd 16 H G nd nd nd nd 8 4. M nd nd 45 L N K In Eercises 5 10, find the vlue of. (See Emple.) Q 5 W Y T X 4 Z In Eercises 19 6, find the vlue of the vrile. (See Emple 4.) y S 16 R 1 y E H F G D ft 1.8 ft 5.8 ft 4.6 ft 3.5 ft 5. z ft 48 hpter 9 Right Tringles nd Trigonometry

24 ERROR NLYSIS In Eercises 7 nd 8, descrie nd correct the error in writing n eqution for the given digrm. 7. y z MTHEMTIL ONNETIONS In Eercises 31 34, find the vlue(s) of the vrile(s) w v z = w (w + v) e g f d 33. y 1 z RESONING Use the digrm. Decide which proportions re true. Select ll tht pply. z 3 4 y h d = f h D MODELING WITH MTHEMTIS In Eercises 9 nd 30, use the digrm. (See Emple 5.) D D = D D = D = D D = D 7. ft 5.5 ft 6 ft 9.5 ft E. 9 E You wnt to determine the height of monument t locl prk. You use crdord squre to line up the top nd ottom of the monument, s shown t the ove left. Your friend mesures the verticl distnce from the ground to your eye nd the horizontl distnce from you to the monument. pproimte the height of the monument. 30. Your clssmte is stnding on the other side of the monument. She hs piece of rope stked t the se of the monument. She etends the rope to the crdord squre she is holding lined up to the top nd ottom of the monument. Use the informtion in the digrm ove to pproimte the height of the monument. Do you get the sme nswer s in Eercise 9? Eplin your resoning. 36. NLYZING RELTIONSHIPS You re designing dimond-shped kite. You know tht D = 44.8 centimeters, D = 7 centimeters, nd = 84.8 centimeters. You wnt to use stright crossr D. out how long should it e? Eplin your resoning. 37. NLYZING RELTIONSHIPS Use the Geometric Men Theorems (Theorems 9.7 nd 9.8) to find nd D. 0 D D 15 Section 9.3 Similr Right Tringles 483

25 38. HOW DO YOU SEE IT? In which of the following tringles does the Geometric Men (ltitude) Theorem (Theorem 9.7) pply? 40. MKING N RGUMENT Your friend clims the geometric men of 4 nd 9 is 6, nd then lels the tringle, s shown. Is your friend correct? Eplin 9 4 your resoning. 6 D In Eercises 41 nd 4, use the given sttements to prove the theorem. Given is right tringle. ltitude D is drwn to hypotenuse. 41. PROVING THEOREM Prove the Geometric Men (ltitude) Theorem (Theorem 9.7) y showing tht D = D D. 39. PROVING THEOREM Use the digrm of. opy nd complete the proof of the Pythgoren Theorem (Theorem 9.1). Given In, is right ngle. Prove c = + STTEMENTS 1. In, is right ngle.. Drw perpendiculr segment (ltitude) from to. RESONS 1.. Perpendiculr Postulte (Postulte 3.) 3. ce = nd cf = ce + = + 4. ddition Property of Equlity 5. ce + cf = c(e + f ) = e + f = 7. Segment ddition Postulte (Postulte 1.) 8. c c = c = + 9. Simplify. Mintining Mthemticl Proficiency Solve the eqution for. (Skills Review Hndook) = 5 f D = 4 c e 4. PROVING THEOREM Prove the Geometric Men (Leg) Theorem (Theorem 9.8) y showing tht = D nd = D. 43. RITIL THINKING Drw right isosceles tringle nd lel the two leg lengths. Then drw the ltitude to the hypotenuse nd lel its length y. Now, use the Right Tringle Similrity Theorem (Theorem 9.6) to drw the three similr tringles from the imge nd lel ny side length tht is equl to either or y. Wht cn you conclude out the reltionship etween the two smller tringles? Eplin your resoning. 44. THOUGHT PROVOKING The rithmetic men nd geometric men of two nonnegtive numers nd y re shown. rithmetic men = + y geometric men = y Write n inequlity tht reltes these two mens. Justify your nswer. 45. PROVING THEOREM Prove the Right Tringle Similrity Theorem (Theorem 9.6) y proving three similrity sttements = 78 Given is right tringle. ltitude D is drwn to hypotenuse. Prove D, D, D D Reviewing wht you lerned in previous grdes nd lessons = hpter 9 Right Tringles nd Trigonometry

26 Wht Did You Lern? ore Voculry Pythgoren triple, p. 464 geometric men, p. 480 ore oncepts Section 9.1 Theorem 9.1 Pythgoren Theorem, p. 464 ommon Pythgoren Triples nd Some of Their Multiples, p. 464 Theorem 9. onverse of the Pythgoren Theorem, p. 466 Theorem 9.3 Pythgoren Inequlities Theorem, p. 467 Section 9. Theorem Tringle Theorem, p. 47 Theorem Tringle Theorem, p. 473 Section 9.3 Theorem 9.6 Right Tringle Similrity Theorem, p. 478 Theorem 9.7 Geometric Men (ltitude) Theorem, p. 480 Theorem 9.8 Geometric Men (Leg) Theorem, p. 480 Mthemticl Prctices 1. In Eercise 31 on pge 469, descrie the steps you took to find the re of the tringle.. In Eercise 3 on pge 476, cn one of the wys e used to show tht ll tringles re similr? Eplin. 3. Eplin why the Geometric Men (ltitude) Theorem (Theorem 9.7) does not pply to three of the tringles in Eercise 38 on pge 484. Study Skills Form Weekly Study Group, Set Up Rules onsider using the following rules. Memers must ttend regulrly, e on time, nd prticipte. The sessions will focus on the key mth concepts, not on the needs of one student. Students who skip clsses will not e llowed to prticipte in the study group. Students who keep the group from eing productive will e sked to leve the group. 485

27 Quiz Find the vlue of. Tell whether the side lengths form Pythgoren triple. (Section 9.1) Verify tht the segment lengths form tringle. Is the tringle cute, right, or otuse? (Section 9.1) 4. 4, 3, nd , 9, nd , 15, nd 10 3 Find the vlues of nd y. Write your nswers in simplest form. (Section 9.) y 8. 8 y y 60 Find the geometric men of the two numers. (Section 9.3) nd nd nd 6 Identify the similr right tringles. Then find the vlue of the vrile. (Section 9.3) D E F 6 H y 9 G 15. J 18 1 z M K L 16. Television sizes re mesured y the length of their digonl. You wnt to purchse television tht is t lest 40 inches. Should you purchse the television shown? Eplin your resoning. (Section 9.1) 17. Ech tringle shown elow is right tringle. (Sections ). re ny of the tringles specil right tringles? Eplin your resoning.. List ll similr tringles, if ny. c. Find the lengths of the ltitudes of tringles nd. 0.5 in. 36 in D E hpter 9 Right Tringles nd Trigonometry

28 9.4 The Tngent Rtio Essentil Question How is right tringle used to find the tngent of n cute ngle? Is there unique right tringle tht must e used? Let e right tringle with cute. The tngent of (written s tn ) is defined s follows. length of leg opposite tn = length of leg djcent to = djcent opposite lculting Tngent Rtio Work with prtner. Use dynmic geometry softwre.. onstruct, s shown. onstruct segments perpendiculr to to form right tringles tht shre verte nd re similr to with vertices, s shown K L M N O P Q J I H G F E D Smple Points (0, 0) (8, 6) (8, 0) ngle m = TTENDING TO PREISION To e proficient in mth, you need to epress numericl nswers with degree of precision pproprite for the prolem contet.. lculte ech given rtio to complete the tle for the deciml vlue of tn for ech right tringle. Wht cn you conclude? Rtio tn KD D LE E MF F Using lcultor NG G OH H Work with prtner. Use clcultor tht hs tngent key to clculte the tngent of Do you get the sme result s in Eplortion 1? Eplin. PI I QJ J ommunicte Your nswer 3. Repet Eplortion 1 for with vertices (0, 0), (8, 5), nd (8, 0). onstruct the seven perpendiculr segments so tht not ll of them intersect t integer vlues of. Discuss your results. 4. How is right tringle used to find the tngent of n cute ngle? Is there unique right tringle tht must e used? Section 9.4 The Tngent Rtio 487

29 9.4 Lesson Wht You Will Lern ore Voculry trigonometric rtio, p. 488 tngent, p. 488 ngle of elevtion, p. 490 REDING Rememer the following revitions. tngent tn opposite opp. djcent dj. Use the tngent rtio. Solve rel-life prolems involving the tngent rtio. Using the Tngent Rtio trigonometric rtio is rtio of the lengths of two sides in right tringle. ll right tringles with given cute ngle re similr y the Similrity Theorem (Theorem 8.3). So, JKL XYZ, nd you cn write KL rewritten s KL JL = YZ XZ YZ = JL. This cn e XZ, which is trigonometric rtio. So, trigonometric rtios re constnt for given ngle mesure. The tngent rtio is trigonometric rtio for cute ngles tht involves the lengths of the legs of right tringle. ore oncept Tngent Rtio Let e right tringle with cute. The tngent of (written s tn ) is defined s follows. length of leg opposite tn = length of leg djcent to = leg opposite J Y Z K L hypotenuse leg djcent to X TTENDING TO PREISION Unless told otherwise, you should round the vlues of trigonometric rtios to four deciml plces nd round lengths to the nerest tenth. In the right tringle ove, nd re complementry. So, is cute. You cn use the sme digrm to find the tngent of. Notice tht the leg djcent to is the leg opposite nd the leg opposite is the leg djcent to. Finding Tngent Rtios Find tn S nd tn R. Write ech nswer s S frction nd s deciml rounded to four plces. 18 T opp. S tn S = dj. to S = RT ST = = opp. R tn R = dj. to R = ST RT = = 9 40 = R Monitoring Progress Help in English nd Spnish t igidesmth.com Find tn J nd tn K. Write ech nswer s frction nd s deciml rounded to four plces. 1. K. L 15 J J 3 L K 488 hpter 9 Right Tringles nd Trigonometry

30 Finding Leg Length Find the vlue of. Round your nswer to the nerest tenth. USING TOOLS STRTEGILLY You cn lso use the Tle of Trigonometric Rtios ville t igidesmth.com to find the deciml pproimtions of trigonometric rtios. Use the tngent of n cute ngle to find leg length. tn 3 = opp. Write rtio for tngent of 3. dj. tn 3 = 11 Sustitute. tn 3 = 11 Multiply ech side y. 11 = tn Divide ech side y tn 3. Use clcultor The vlue of is out STUDY TIP The tngents of ll 60 ngles re the sme constnt rtio. ny right tringle with 60 ngle cn e used to determine this vlue. You cn find the tngent of n cute ngle mesuring 30, 45, or 60 y pplying wht you know out specil right tringles. Using Specil Right Tringle to Find Tngent Use specil right tringle to find the tngent of 60 ngle. Step 1 ecuse ll tringles re similr, you cn simplify your clcultions y choosing 1 s the length of the shorter leg. Use the Tringle Theorem (Theorem 9.5) to find the length of the longer leg. longer leg = shorter leg Tringle Theorem = 1 3 Sustitute. = 3 Simplify Step Find tn 60. tn 60 = opp. dj. tn 60 = 3 1 tn 60 = 3 3 Write rtio for tngent of 60. Sustitute. Simplify. The tngent of ny 60 ngle is Monitoring Progress Help in English nd Spnish t igidesmth.com Find the vlue of. Round your nswer to the nerest tenth WHT IF? In Emple 3, the length of the shorter leg is 5 insted of 1. Show tht the tngent of 60 is still equl to 3. Section 9.4 The Tngent Rtio 489

31 Solving Rel-Life Prolems The ngle tht n upwrd line of sight mkes with horizontl line is clled the ngle of elevtion. Modeling with Mthemtics You re mesuring the height of spruce tree. You stnd 45 feet from the se of the tree. You mesure the ngle of elevtion from the ground to the top of the tree to e 59. Find the height h of the tree to the nerest foot. h ft ft 1. Understnd the Prolem You re given the ngle of elevtion nd the distnce from the tree. You need to find the height of the tree to the nerest foot.. Mke Pln Write trigonometric rtio for the tngent of the ngle of elevtion involving the height h. Then solve for h. 3. Solve the Prolem opp. tn 59 = dj. h tn 59 = tn 59 = h Write rtio for tngent of 59. Sustitute. Multiply ech side y h Use clcultor. The tree is out 75 feet tll. 4. Look ck heck your nswer. ecuse 59 is close to 60, the vlue of h should e close to the length of the longer leg of tringle, where the length of the shorter leg is 45 feet. longer leg = shorter leg 3 = Tringle Theorem Sustitute. Use clcultor. The vlue of 77.9 feet is close to the vlue of h. Monitoring Progress h in. Help in English nd Spnish t igidesmth.com 6. You re mesuring the height of lmppost. You stnd 40 inches from the se of in. 490 hpter 9 hs_geo_pe_0904.indd 490 the lmppost. You mesure the ngle of elevtion from the ground to the top of the lmppost to e 70. Find the height h of the lmppost to the nerest inch. Right Tringles nd Trigonometry 1/19/15 1:43 PM

32 9.4 Eercises Dynmic Solutions ville t igidesmth.com Voculry nd ore oncept heck 1. OMPLETE THE SENTENE The tngent rtio compres the length of to the length of.. WRITING Eplin how you know the tngent rtio is constnt for given ngle mesure. Monitoring Progress nd Modeling with Mthemtics In Eercises 3 6, find the tngents of the cute ngles in the right tringle. Write ech nswer s frction nd s deciml rounded to four deciml plces. (See Emple 1.) 3. R 8 T 5. G 1 J 5 H S 4. E 7 4 F 5 D 6. J L 3 5 In Eercises 7 10, find the vlue of. Round your nswer to the nerest tenth. (See Emple.) ERROR NLYSIS In Eercises 11 nd 1, descrie the error in the sttement of the tngent rtio. orrect the error if possile. Otherwise, write not possile. 6 K tn 55 = 11.0 In Eercises 13 nd 14, use specil right tringle to find the tngent of the given ngle mesure. (See Emple 3.) MODELING WITH MTHEMTIS surveyor is stnding 118 feet from the se of the Wshington Monument. The surveyor mesures the ngle of elevtion from the ground to the top of the monument to e 78. Find the height h of the Wshington Monument to the nerest foot. (See Emple 4.) 16. MODELING WITH MTHEMTIS Scientists cn mesure the depths of crters on the moon y looking t photos of shdows. The length of the shdow cst y the edge of crter is 500 meters. The ngle of elevtion of the rys of the Sun is 55. Estimte the depth d of the crter. Sun s ry D 37 1 E 35 F tn D = m 17. USING STRUTURE Find the tngent of the smller cute ngle in right tringle with side lengths 5, 1, nd 13. d Section 9.4 The Tngent Rtio 491

33 18. USING STRUTURE Find the tngent of the lrger cute ngle in right tringle with side lengths 3, 4, nd RESONING How does the tngent of n cute ngle in right tringle chnge s the ngle mesure increses? Justify your nswer. 0. RITIL THINKING For wht ngle mesure(s) is the tngent of n cute ngle in right tringle equl to 1? greter thn 1? less thn 1? Justify your nswer. 1. MKING N RGUMENT Your fmily room hs sliding-glss door. You wnt to uy n wning for the door tht will e just long enough to keep the Sun out when it is t its highest point in the sky. The ngle of elevtion of the rys of the Sun t this point is 70, nd the height of the door is 8 feet. Your sister clims you cn determine how fr the overhng should etend y multiplying 8 y tn 70. Is your sister correct? Eplin. 4. THOUGHT PROVOKING To crete the digrm elow, you egin with n isosceles right tringle with legs 1 unit long. Then the hypotenuse of the first tringle ecomes the leg of second tringle, whose remining leg is 1 unit long. ontinue the digrm until you hve constructed n ngle whose tngent is 1. pproimte the mesure of this ngle PROLEM SOLVING Your clss is hving clss picture tken on the lwn. The photogrpher is positioned 14 feet wy from the center of the clss. The photogrpher turns 50 to look t either end of the clss. 1 1 Sun s ry 8 ft ft HOW DO YOU SEE IT? Write epressions for the tngent of ech cute ngle in the right tringle. Eplin how the tngent of one cute ngle is relted to the tngent of the other cute ngle. Wht kind of ngle pir is nd? 3. RESONING Eplin why it is not possile to find the tngent of right ngle or n otuse ngle. c Mintining Mthemticl Proficiency Find the vlue of. (Section 9.) Wht is the distnce etween the ends of the clss?. The photogrpher turns nother 10 either wy to see the end of the cmer rnge. If ech student needs feet of spce, out how mny more students cn fit t the end of ech row? Eplin. 6. PROLEM SOLVING Find the perimeter of the figure, where = 6, D = F, nd D is the midpoint of. E 50 D 35 G Reviewing wht you lerned in previous grdes nd lessons F H 49 hpter 9 Right Tringles nd Trigonometry

34 9.5 The Sine nd osine Rtios Essentil Question How is right tringle used to find the sine nd cosine of n cute ngle? Is there unique right tringle tht must e used? Let e right tringle with cute. The sine of nd cosine of (written s sin nd cos, respectively) re defined s follows. length of leg opposite sin = length of hypotenuse = hypotenuse opposite length of leg djcent to cos = length of hypotenuse = djcent lculting Sine nd osine Rtios Work with prtner. Use dynmic geometry softwre.. onstruct, s shown. onstruct segments perpendiculr to to form right tringles tht shre verte nd re similr to with vertices, s shown K L M N O P Q J I H G F E D Smple Points (0, 0) (8, 6) (8, 0) ngle m = lculte ech given rtio to complete the tle for the deciml vlues of sin nd cos for ech right tringle. Wht cn you conclude? Sine rtio KD K LE L MF M NG N OH O PI P QJ Q sin osine rtio D K E L F M G N H O I P J Q LOOKING FOR STRUTURE To e proficient in mth, you need to look closely to discern pttern or structure. cos ommunicte Your nswer. How is right tringle used to find the sine nd cosine of n cute ngle? Is there unique right tringle tht must e used? 3. In Eplortion 1, wht is the reltionship etween nd in terms of their mesures? Find sin nd cos. How re these two vlues relted to sin nd cos? Eplin why these reltionships eist. Section 9.5 The Sine nd osine Rtios 493

35 9.5 Lesson Wht You Will Lern ore Voculry sine, p. 494 cosine, p. 494 ngle of depression, p. 497 Use the sine nd cosine rtios. Find the sine nd cosine of ngle mesures in specil right tringles. Solve rel-life prolems involving sine nd cosine rtios. Using the Sine nd osine Rtios The sine nd cosine rtios re trigonometric rtios for cute ngles tht involve the lengths of leg nd the hypotenuse of right tringle. ore oncept REDING Rememer the following revitions. sine sin cosine cos hypotenuse hyp. Sine nd osine Rtios Let e right tringle with cute. The sine of nd cosine of (written s sin nd cos ) re defined s follows. length of leg opposite sin = length of hypotenuse length of leg djcent to cos = length of hypotenuse = = leg opposite hypotenuse leg djcent to Finding Sine nd osine Rtios Find sin S, sin R, cos S, nd cos R. Write ech nswer s frction nd s deciml rounded to four plces. opp. S sin S = hyp. = RT dj. to S cos S = = ST hyp. SR = SR = 63 opp. R sin R = = ST 65 hyp. SR = cos R = dj. to R hyp. R = RT S 16 T SR = In Emple 1, notice tht sin S = cos R nd sin R = cos S. This is true ecuse the side opposite S is djcent to R nd the side opposite R is djcent to S. The reltionship etween the sine nd cosine of S nd R is true for ll complementry ngles. ore oncept Sine nd osine of omplementry ngles The sine of n cute ngle is equl to the cosine of its complement. The cosine of n cute ngle is equl to the sine of its complement. Let nd e complementry ngles. Then the following sttements re true. sin = cos(90 ) = cos sin = cos(90 ) = cos cos = sin(90 ) = sin cos = sin(90 ) = sin 494 hpter 9 Right Tringles nd Trigonometry

36 Rewriting Trigonometric Epressions Write sin 56 in terms of cosine. Use the fct tht the sine of n cute ngle is equl to the cosine of its complement. sin 56 = cos(90 56 ) = cos 34 The sine of 56 is the sme s the cosine of 34. You cn use the sine nd cosine rtios to find unknown mesures in right tringles. Finding Leg Lengths Find the vlues of nd y using sine nd cosine. Round your nswers to the nerest tenth. 6 Step 1 Use sine rtio to find the vlue of. y sin 6 = opp. Write rtio for sine of 6. hyp. 14 sin 6 = Sustitute sin 6 = Multiply ech side y Use clcultor. The vlue of is out 6.1. Step Use cosine rtio to find the vlue of y. cos 6 = dj. hyp. Write rtio for cosine of 6. cos 6 = y Sustitute cos 6 = y Multiply ech side y y Use clcultor. The vlue of y is out 1.6. Monitoring Progress Help in English nd Spnish t igidesmth.com 1. Find sin D, sin F, cos D, nd cos F. Write ech nswer s frction nd s deciml rounded to four plces. 4 E 7 F 5. Write cos 3 in terms of sine. D 3. Find the vlues of u nd t using sine nd cosine. Round your nswers to the nerest tenth. u 65 8 t Section 9.5 The Sine nd osine Rtios 495

37 Finding Sine nd osine in Specil Right Tringles Find the sine nd cosine of 45 ngle. Finding the Sine nd osine of 45 egin y sketching tringle. ecuse ll such tringles re similr, you cn simplify your clcultions y choosing 1 s the length of ech leg. Using the Tringle Theorem (Theorem 9.4), the length of the hypotenuse is. STUDY TIP Notice tht sin 45 = cos(90 45) = cos 45. sin 45 = opp. hyp cos 45 = dj. hyp. = 1 = = 1 = Find the sine nd cosine of 30 ngle. Finding the Sine nd osine of 30 egin y sketching tringle. ecuse ll such tringles re similr, you cn simplify your clcultions y choosing 1 s the length of the shorter leg. Using the Tringle Theorem (Theorem 9.5), the length of the longer leg is 3 nd the length of the hypotenuse is sin 30 = opp. hyp. cos 30 = dj. hyp. = 1 = 3 = Monitoring Progress Help in English nd Spnish t igidesmth.com 4. Find the sine nd cosine of 60 ngle. 496 hpter 9 Right Tringles nd Trigonometry

38 Solving Rel-Life Prolems Recll from the previous lesson tht the ngle n upwrd line of sight mkes with horizontl line is clled the ngle of elevtion. The ngle tht downwrd line of sight mkes with horizontl line is clled the ngle of depression. Modeling with Mthemtics You re skiing on mountin with n ltitude of 100 feet. The ngle of depression is 1. Find the distnce you ski down the mountin to the nerest foot. 1 ft 100 ft Not drwn to scle 1. Understnd the Prolem You re given the ngle of depression nd the ltitude of the mountin. You need to find the distnce tht you ski down the mountin.. Mke Pln Write trigonometric rtio for the sine of the ngle of depression involving the distnce. Then solve for. 3. Solve the Prolem sin 1 = opp. Write rtio for sine of 1. hyp. sin 1 = 100 Sustitute. sin 1 = 100 Multiply ech side y. = 100 sin Divide ech side y sin 1. Use clcultor. You ski out 3349 feet down the mountin. 4. Look ck heck your nswer. The vlue of sin 1 is out Sustitute for in the sine rtio nd compre the vlues This vlue is pproimtely the sme s the vlue of sin 1. Monitoring Progress Help in English nd Spnish t igidesmth.com 5. WHT IF? In Emple 6, the ngle of depression is 8. Find the distnce you ski down the mountin to the nerest foot. Section 9.5 The Sine nd osine Rtios 497

39 9.5 Eercises Dynmic Solutions ville t igidesmth.com Voculry nd ore oncept heck 1. VOULRY The sine rtio compres the length of to the length of.. WHIH ONE DOESN T ELONG? Which rtio does not elong with the other three? Eplin your resoning. sin cos tn Monitoring Progress nd Modeling with Mthemtics In Eercises 3 8, find sin D, sin E, cos D, nd cos E. Write ech nswer s frction nd s deciml rounded to four plces. (See Emple 1.) In Eercises 17, find the vlue of ech vrile using sine nd cosine. Round your nswers to the nerest tenth. (See Emple 3.) 3. 9 F D q D 15 E E 35 F y 3 p 5. D 45 F 6. D 19. v E E F w r s 7. D 6 E F 8. D F E m 50 n 8 In Eercises 9 1, write the epression in terms of cosine. (See Emple.) 9. sin sin sin 9 1. sin 64 In Eercises 13 16, write the epression in terms of sine. 13. cos cos 4 3. RESONING Which rtios re equl? Select ll tht pply. (See Emple 4.) X 3 45 Y 3 sin X cos X sin Z cos Z 3 Z 15. cos cos hpter 9 Right Tringles nd Trigonometry

40 4. RESONING Which rtios re equl to 1? Select ll tht pply. (See Emple 5.) J K sin L cos L sin J cos J 5. ERROR NLYSIS Descrie nd correct the error in finding sin L sin = PROLEM SOLVING You re flying kite with 0 feet of string etended. The ngle of elevtion from the spool of string to the kite is 67.. Drw nd lel digrm tht represents the sitution.. How fr off the ground is the kite if you hold the spool 5 feet off the ground? Descrie how the height where you hold the spool ffects the height of the kite. 30. MODELING WITH MTHEMTIS Plnes tht fly t high speeds nd low elevtions hve rdr systems tht cn determine the rnge of n ostcle nd the ngle of elevtion to the top of the ostcle. The rdr of plne flying t n ltitude of 0,000 feet detects tower tht is 5,000 feet wy, with n ngle of elevtion of 1. 5,000 ft 1 h ft 6. WRITING Eplin how to tell which side of right tringle is djcent to n ngle nd which side is the hypotenuse. 7. MODELING WITH MTHEMTIS The top of the slide is 1 feet from the ground nd hs n ngle of depression of 53. Wht is the length of the slide? (See Emple 6.) Not drwn to scle. How mny feet must the plne rise to pss over the tower?. Plnes cnnot come closer thn 1000 feet verticlly to ny oject. t wht ltitude must the plne fly in order to pss over the tower? 1 ft MKING N RGUMENT Your friend uses the eqution sin 49 = to find. Your cousin uses 16 the eqution cos 41 = to find. Who is correct? 16 Eplin your resoning y 8. MODELING WITH MTHEMTIS Find the horizontl distnce the escltor covers. ft 41 6 ft 3. WRITING Descrie wht you must know out tringle in order to use the sine rtio nd wht you must know out tringle in order to use the cosine rtio. 33. MTHEMTIL ONNETIONS If EQU is equilterl nd RGT is right tringle with RG =, RT = 1, nd m T = 90, show tht sin E = cos G. Section 9.5 The Sine nd osine Rtios 499

41 34. MODELING WITH MTHEMTIS Sumrines use sonr systems, which re similr to rdr systems, to detect ostcles. Sonr systems use sound to detect ojects under wter m Not drwn to scle m 37. MULTIPLE REPRESENTTIONS You re stnding on cliff ove n ocen. You see silot from your vntge point 30 feet ove the ocen.. Drw nd lel digrm of the sitution.. Mke tle showing the ngle of depression nd the length of your line of sight. Use the ngles 40, 50, 60, 70, nd 80. c. Grph the vlues you found in prt (), with the ngle mesures on the -is. d. Predict the length of the line of sight when the ngle of depression is 30.. You re trveling underwter in sumrine. The sonr system detects n iceerg 4000 meters hed, with n ngle of depression of 34 to the ottom of the iceerg. How mny meters must the sumrine lower to pss under the iceerg?. The sonr system then detects sunken ship 1500 meters hed, with n ngle of elevtion of 19 to the highest prt of the sunken ship. How mny meters must the sumrine rise to pss over the sunken ship? 35. STRT RESONING Mke conjecture out how you could use trigonometric rtios to find ngle mesures in tringle. 36. HOW DO YOU SEE IT? Using only the given informtion, would you use sine rtio or cosine rtio to find the length of the hypotenuse? Eplin your resoning THOUGHT PROVOKING One of the following infinite series represents sin nd the other one represents cos (where is mesured in rdins). Which is which? Justify your nswer. Then use ech series to pproimte the sine nd cosine of π 6. (Hints: π = 180 ; 5! = ; Find the vlues tht the sine nd cosine rtios pproch s the ngle mesure pproches zero.). 3 3! + 5 5! 7 7! ! + 4 4! 6 6! RITIL THINKING Let e ny cute ngle of right tringle. Show tht () tn = sin cos nd () (sin ) + (cos ) = RITIL THINKING Eplin why the re of in the digrm cn e found using the formul re = 1 sin. Then clculte the re when = 4, = 7, nd m = 40. h c Mintining Mthemticl Proficiency Find the vlue of. Tell whether the side lengths form Pythgoren triple. (Section 9.1) Reviewing wht you lerned in previous grdes nd lessons hpter 9 Right Tringles nd Trigonometry

42 9.6 Solving Right Tringles Essentil Question When you know the lengths of the sides of right tringle, how cn you find the mesures of the two cute ngles? Solving Specil Right Tringles Work with prtner. Use the figures to find the vlues of the sine nd cosine of nd. Use these vlues to find the mesures of nd. Use dynmic geometry softwre to verify your nswers TTENDING TO PREISION To e proficient in mth, you need to clculte ccurtely nd efficiently, epressing numericl nswers with degree of precision pproprite for the prolem contet Solving Right Tringles Work with prtner. You cn use clcultor to find the mesure of n ngle when you know the vlue of the sine, cosine, or tngent of the ngle. Use the inverse sine, inverse cosine, or inverse tngent feture of your clcultor to pproimte the mesures of nd to the nerest tenth of degree. Then use dynmic geometry softwre to verify your nswers ommunicte Your nswer 3. When you know the lengths of the sides of right tringle, how cn you find the mesures of the two cute ngles? 4. ldder lening ginst uilding forms right tringle with the uilding nd the ground. The legs of the right tringle (in meters) form Pythgoren triple. Find the mesures of the two cute ngles to the nerest tenth of degree. Section 9.6 Solving Right Tringles 501

43 9.6 Lesson Wht You Will Lern ore Voculry inverse tngent, p. 50 inverse sine, p. 50 inverse cosine, p.50 solve right tringle, p. 503 Use inverse trigonometric rtios. Solve right tringles. Using Inverse Trigonometric Rtios Identifying ngles from Trigonometric Rtios Determine which of the two cute ngles hs cosine of 0.5. Find the cosine of ech cute ngle. 3 dj. to cos = = 3 dj. to cos = = 1 hyp. hyp. = The cute ngle tht hs cosine of 0.5 is. If the mesure of n cute ngle is 60, then its cosine is 0.5. The converse is lso true. If the cosine of n cute ngle is 0.5, then the mesure of the ngle is 60. So, in Emple 1, the mesure of must e 60 ecuse its cosine is 0.5. REDING The epression tn 1 is red s the inverse tngent of. NOTHER WY You cn use the Tle of Trigonometric Rtios ville t igidesmth.com to pproimte tn to the nerest degree. Find the numer closest to 0.75 in the tngent column nd red the ngle mesure t the left. ore oncept Inverse Trigonometric Rtios Let e n cute ngle. Inverse Tngent If tn =, then tn 1 = m. Finding ngle Mesures Let,, nd e cute ngles. Use clcultor to pproimte the mesures of,, nd to the nerest tenth of degree.. tn = sin = 0.87 c. cos = m = tn m = sin c. m = cos tn 1 1 = m Inverse Sine If sin = y, then sin 1 y = m. sin = m Inverse osine If cos = z, then cos 1 z = m. cos 1 = m F 13 1 E 5 D Monitoring Progress Help in English nd Spnish t igidesmth.com Determine which of the two cute ngles hs the given trigonometric rtio. 1. The sine of the ngle is The tngent of the ngle is hpter 9 Right Tringles nd Trigonometry

44 Monitoring Progress Solving Right Tringles ore oncept Solving Right Tringle Help in English nd Spnish t igidesmth.com Let G, H, nd K e cute ngles. Use clcultor to pproimte the mesures of G, H, nd K to the nerest tenth of degree. 3. tn G = sin H = cos K = 0.94 To solve right tringle mens to find ll unknown side lengths nd ngle mesures. You cn solve right tringle when you know either of the following. two side lengths one side length nd the mesure of one cute ngle Solving Right Tringle Solve the right tringle. Round deciml nswers to the nerest tenth. 3 NOTHER WY You could lso hve found m first y finding tn Step 1 Use the Pythgoren Theorem (Theorem 9.1) to find the length of the hypotenuse. c = + Pythgoren Theorem c = 3 + c = 13 c = 13 c 3.6 Sustitute. Simplify. Find the positive squre root. Use clcultor. Step Find m. m = tn Use clcultor. 3 Step 3 Find m. ecuse nd re complements, you cn write m = 90 m = In, c 3.6, m 33.7, nd m c Monitoring Progress Help in English nd Spnish t igidesmth.com Solve the right tringle. Round deciml nswers to the nerest tenth. 6. D 7. G F 1 E H 91 J Section 9.6 Solving Right Tringles 503

45 Solving Right Tringle Solve the right tringle. Round deciml nswers to the nerest tenth. H g 5 Use trigonometric rtios to find the vlues of g nd h. sin H = opp. hyp. 13 J h G cos H = dj. hyp. REDING rked stge slnts upwrd from front to ck to give the udience etter view. sin 5 = h 13 cos 5 = g sin 5 = h 13 cos 5 = g 5.5 h 11.8 g ecuse H nd G re complements, you cn write m G = 90 m H = 90 5 = 65. In GHJ, h 5.5, g 11.8, nd m G = 65. Solving Rel-Life Prolem Your school is uilding rked stge. The stge will e 30 feet long from front to ck, with totl rise of feet. You wnt the rke (ngle of elevtion) to e 5 or less for sfety. Is the rked stge within your desired rnge? 30 ft stge ck ft stge front Use the inverse sine rtio to find the degree mesure of the rke. sin The rke is out 3.8, so it is within your desired rnge of 5 or less. Monitoring Progress 8. Solve the right tringle. Round deciml nswers to the nerest tenth. 9. WHT IF? In Emple 5, suppose nother rked stge is 0 feet long from front to ck with totl rise of feet. Is the rked stge within your desired rnge? Help in English nd Spnish t igidesmth.com Y X Z 504 hpter 9 Right Tringles nd Trigonometry

46 9.6 Eercises Dynmic Solutions ville t igidesmth.com Voculry nd ore oncept heck 1. OMPLETE THE SENTENE To solve right tringle mens to find the mesures of ll its nd.. WRITING Eplin when you cn use trigonometric rtio to find side length of right tringle nd when you cn use the Pythgoren Theorem (Theorem 9.1). Monitoring Progress nd Modeling with Mthemtics In Eercises 3 6, determine which of the two cute ngles hs the given trigonometric rtio. (See Emple 1.) 3. The cosine of the 4. The sine of the ngle is 4 5. ngle is The sine of the 6. The tngent of the ngle is ngle is In Eercises 7 1, let D e n cute ngle. Use clcultor to pproimte the mesure of D to the nerest tenth of degree. (See Emple.) 7. sin D = sin D = cos D = cos D = tn D = tn D = 0.7 In Eercises 13 18, solve the right tringle. Round deciml nswers to the nerest tenth. (See Emples 3 nd 4.) E D 15. Y Z 16. G 14 H X M 8 40 K L 18. R S ERROR NLYSIS Descrie nd correct the error in using n inverse trigonometric rtio. T V 8 U T sin = m T 0. PROLEM SOLVING In order to unlod cly esily, the ody of dump truck must e elevted to t lest 45. The ody of dump truck tht is 14 feet long hs een rised 8 feet. Will the cly pour out esily? Eplin your resoning. (See Emple 5.) 1. PROLEM SOLVING You re stnding on footridge tht is 1 feet ove lke. You look down nd see duck in the wter. The duck is 7 feet wy from the footridge. Wht is the ngle of elevtion from the duck to you? 1 ft J F 7 ft Section 9.6 Solving Right Tringles 505

47 . HOW DO YOU SEE IT? Write three epressions tht cn e used to pproimte the mesure of. Which epression would you choose? Eplin your choice MODELING WITH MTHEMTIS The Uniform Federl ccessiility Stndrds specify tht wheelchir rmp my not hve n incline greter thn You wnt to uild rmp with verticl rise of 8 inches. You wnt to minimize the horizontl distnce tken up y the rmp. Drw digrm showing the pproimte dimensions of your rmp. 4. MODELING WITH MTHEMTIS The horizontl prt of step is clled the tred. The verticl prt is clled the riser. The recommended riser-to-tred rtio is 7 inches : 11 inches.. Find the vlue of for stirs uilt using the recommended riser-to-tred rtio. tred riser. You wnt to uild stirs tht re less steep thn the stirs in prt (). Give n emple of riser-totred rtio tht you could use. Find the vlue of for your stirs. 5. USING TOOLS Find the mesure of R without using protrctor. Justify your technique. Q 6. MKING N RGUMENT Your friend clims tht tn 1 1 =. Is your friend correct? Eplin your tn resoning. USING STRUTURE In Eercises 7 nd 8, solve ech tringle. 7. JKM nd LKM J 41 9 ft K M 8. TUS nd VTW T U 4 m 5 m 64 S 1 ft V 9 m W 9. MTHEMTIL ONNETIONS Write n epression tht cn e used to find the mesure of the cute ngle formed y ech line nd the -is. Then pproimte the ngle mesure to the nerest tenth of degree.. y = 3. y = THOUGHT PROVOKING Simplify ech epression. Justify your nswer.. sin 1 (sin ). tn(tn 1 y) c. cos(cos 1 z) 31. RESONING Eplin why the epression sin 1 (1.) does not mke sense. L P Mintining Mthemticl Proficiency Solve the eqution. (Skills Review Hndook) = = 18 R USING STRUTURE The perimeter of rectngle D is 16 centimeters, nd the rtio of its width to its length is 1 : 3. Segment D divides the rectngle into two congruent tringles. Find the side lengths nd ngle mesures of these two tringles. Reviewing wht you lerned in previous grdes nd lessons.1 = = hpter 9 Right Tringles nd Trigonometry

48 9.7 Lw of Sines nd Lw of osines Essentil Question Wht re the Lw of Sines nd the Lw of osines? Discovering the Lw of Sines Work with prtner.. opy nd complete the tle for the tringle shown. Wht cn you conclude? c Smple Segments = 3.16 = 6.3 c = 5.10 ngles m = 9.74 m = m = USING TOOLS STRTEGILLY To e proficient in mth, you need to use technology to compre predictions with dt. m sin m sin m c sin c. Use dynmic geometry softwre to drw two other tringles. opy nd complete the tle in prt () for ech tringle. Use your results to write conjecture out the reltionship etween the sines of the ngles nd the lengths of the sides of tringle. Discovering the Lw of osines Work with prtner.. opy nd complete the tle for the tringle in Eplortion 1(). Wht cn you conclude? c c m + cos. Use dynmic geometry softwre to drw two other tringles. opy nd complete the tle in prt () for ech tringle. Use your results to write conjecture out wht you oserve in the completed tles. ommunicte Your nswer 3. Wht re the Lw of Sines nd the Lw of osines? 4. When would you use the Lw of Sines to solve tringle? When would you use the Lw of osines to solve tringle? Section 9.7 Lw of Sines nd Lw of osines 507

49 9.7 Lesson Wht You Will Lern ore Voculry Lw of Sines, p. 509 Lw of osines, p. 511 Find res of tringles. Use the Lw of Sines to solve tringles. Use the Lw of osines to solve tringles. Finding res of Tringles So fr, you hve used trigonometric rtios to solve right tringles. In this lesson, you will lern how to solve ny tringle. When the tringle is otuse, you my need to find trigonometric rtio for n otuse ngle. Finding Trigonometric Rtios for Otuse ngles Use clcultor to find ech trigonometric rtio. Round your nswer to four deciml plces.. tn 150. sin 10 c. cos 95. tn sin c. cos Monitoring Progress Help in English nd Spnish t igidesmth.com Use clcultor to find the trigonometric rtio. Round your nswer to four deciml plces. 1. tn 110. sin cos 165 ore oncept re of Tringle The re of ny tringle is given y one-hlf the product of the lengths of two sides times the sine of their included ngle. For shown, there re three wys to clculte the re. c re = 1 c sin re = 1 c sin re = 1 sin Finding the re of Tringle Find the re of the tringle. Round your nswer to the nerest tenth. re = 1 c sin = 1 (17)(19) sin The re of the tringle is out 114. squre units. Monitoring Progress Help in English nd Spnish t igidesmth.com Find the re of with the given side lengths nd included ngle. Round your nswer to the nerest tenth. 4. m = 60, = 19, c = m = 9, = 38, = hpter 9 Right Tringles nd Trigonometry

50 Using the Lw of Sines The trigonometric rtios in the previous sections cn only e used to solve right tringles. You will lern two lws tht cn e used to solve ny tringle. You cn use the Lw of Sines to solve tringles when two ngles nd the length of ny side re known (S or S cses), or when the lengths of two sides nd n ngle opposite one of the two sides re known (SS cse). Theorem Theorem 9.9 Lw of Sines The Lw of Sines cn e written in either of the following forms for with sides of length,, nd c. sin = sin = sin c Proof E. 51, p. 516 sin = sin = c sin c Using the Lw of Sines (SS se) Solve the tringle. Round deciml nswers to the nerest tenth. Use the Lw of Sines to find m. sin sin 11 = sin = sin sin 115 sin = 0 m 9.9 Lw of Sines Sustitute. Multiply ech side y 11. Use clcultor. y the Tringle Sum Theorem (Theorem 5.1), m = Use the Lw of Sines gin to find the remining side length c of the tringle. c sin = sin c sin 35.1 = 0 sin sin 35.1 c = sin 115 c 1.7 Lw of Sines Sustitute. Multiply ech side y sin Use clcultor. In, m 9.9, m 35.1, nd c 1.7. Monitoring Progress Help in English nd Spnish t igidesmth.com Solve the tringle. Round deciml nswers to the nerest tenth c Section 9.7 Lw of Sines nd Lw of osines 509

51 Using the Lw of Sines (S se) Solve the tringle. Round deciml nswers to the nerest tenth c y the Tringle Sum Theorem (Theorem 5.1), m = = 48. y the Lw of Sines, you cn write sin 48 = 15 sin 5 = c sin 107. sin 48 = 15 Write two equtions, ech c sin 5 with one vrile. sin 107 = 15 sin 5 15 sin sin 107 = Solve for ech vrile. c = sin 5 sin Use clcultor. c 33.9 In, m = 48, 6.4, nd c surveyor mkes the mesurements shown to determine the length of ridge to e uilt cross smll lke from the North Picnic re to the South Picnic re. Find the length of the ridge. Using the Lw of Sines (S se) South Picnic re In the digrm, c represents the distnce from the North Picnic re to the South Picnic re, so c represents the length of the ridge. y the Tringle Sum Theorem (Theorem 5.1), m = = 49. y the Lw of Sines, you cn write sin 71 = 150 sin 49 = c sin 60. c sin 60 = 150 Write n eqution involving c. sin sin 60 c = sin 49 c 17.1 Multiply ech side y sin 60. Use clcultor. The length of the ridge will e out 17.1 meters. Monitoring Progress Help in English nd Spnish t igidesmth.com Solve the tringle. Round deciml nswers to the nerest tenth North Picnic re WHT IF? In Emple 5, wht would e the length of ridge from the South Picnic re to the Est Picnic re? c m 60 Est Picnic re 510 hpter 9 Right Tringles nd Trigonometry

52 Using the Lw of osines You cn use the Lw of osines to solve tringles when two sides nd the included ngle re known (SS cse), or when ll three sides re known (SSS cse). Theorem Theorem 9.10 Lw of osines If hs sides of length,, nd c, s shown, then the following re true. = + c c cos = + c c cos c = + cos Proof E. 5, p. 516 c Using the Lw of osines (SS se) NOTHER WY When you know ll three sides nd one ngle, you cn use the Lw of osines or the Lw of Sines to find the mesure of second ngle. OMMON ERROR In Emple 6, the smller remining ngle is found first ecuse the inverse sine feture of clcultor only gives ngle mesures from 0 to 90. So, when n ngle is otuse, like ecuse 14 > (7.85) + 11, you will not get the otuse mesure. Solve the tringle. Round deciml nswers to the nerest tenth. Use the Lw of osines to find side length. = + c c cos = (11)(14) cos 34 = cos 34 Lw of osines Sustitute. Simplify. = cos 34 Find the positive squre root. 7.9 Use the Lw of Sines to find m. sin = sin sin 11 = sin cos sin 34 sin = cos 34 m 51.6 Use clcultor. Lw of Sines Sustitute. Multiply ech side y 11. Use clcultor. y the Tringle Sum Theorem (Theorem 5.1), m = In, 7.9, m 51.6, nd m Monitoring Progress Help in English nd Spnish t igidesmth.com Solve the tringle. Round deciml nswers to the nerest tenth Section 9.7 Lw of Sines nd Lw of osines 511

53 Using the Lw of osines (SSS se) OMMON ERROR In Emple 7, the lrgest ngle is found first to mke sure tht the other two ngles re cute. This wy, when you use the Lw of Sines to find nother ngle mesure, you will know tht it is etween 0 nd 90. Solve the tringle. Round deciml nswers to the nerest tenth. First, find the ngle opposite the longest side,. Use the Lw of osines to find m. = + c c cos Lw of osines 7 = (1)(0) cos Sustitute = cos (1)(0) Solve for cos. m 11.7 Use clcultor. Now, use the Lw of Sines to find m. sin = sin sin sin 11.7 = sin 11.7 sin = 7 m 4. Lw of Sines Sustitute for,, nd. Multiply ech side y 1. Use clcultor. y the Tringle Sum Theorem (Theorem 5.1), m = In, m 4., m 11.7, nd m Solving Rel-Life Prolem n orgnism s step ngle is mesure of wlking efficiency. The closer the step ngle is to 180, the more efficiently the orgnism wlked. The digrm shows set of footprints for dinosur. Find the step ngle. 155 cm 197 cm 316 cm = + c c cos Lw of osines 316 = (155)(197) cos Sustitute = cos (155)(197) Solve for cos m Use clcultor. The step ngle is out Monitoring Progress Help in English nd Spnish t igidesmth.com Solve the tringle. Round deciml nswers to the nerest tenth hpter 9 Right Tringles nd Trigonometry

54 9.7 Eercises Dynmic Solutions ville t igidesmth.com Voculry nd ore oncept heck 1. WRITING Wht type of tringle would you use the Lw of Sines or the Lw of osines to solve?. VOULRY Wht informtion do you need to use the Lw of Sines? Monitoring Progress nd Modeling with Mthemtics In Eercises 3 8, use clcultor to find the trigonometric rtio. Round your nswer to four deciml plces. (See Emple 1.) 3. sin sin cos cos tn tn 116 In Eercises 9 1, find the re of the tringle. Round your nswer to the nerest tenth. (See Emple.) In Eercises 13 18, solve the tringle. Round deciml nswers to the nerest tenth. (See Emples 3, 4, nd 5.) In Eercises 19 4, solve the tringle. Round deciml nswers to the nerest tenth. (See Emples 6 nd 7.) ERROR NLYSIS Descrie nd correct the error in finding m sin sin 55 = sin 55 sin = 5 m 79.4 Section 9.7 Lw of Sines nd Lw of osines 513

55 6. ERROR NLYSIS Descrie nd correct the error in finding m in when = 19, = 1, nd c = 11. cos = (19)(1) m 75.4 OMPRING METHODS In Eercises 7 3, tell whether you would use the Lw of Sines, the Lw of osines, or the Pythgoren Theorem (Theorem 9.1) nd trigonometric rtios to solve the tringle with the given informtion. Eplin your resoning. Then solve the tringle. 35. MODELING WITH MTHEMTIS You re on the oservtion deck of the Empire Stte uilding looking t the hrysler uilding. When you turn 145 clockwise, you see the Sttue of Lierty. You know tht the hrysler uilding nd the Empire Stte uilding re out 0.6 mile prt nd tht the hrysler uilding nd the Sttue of Lierty re out 5.6 miles prt. Estimte the distnce etween the Empire Stte uilding nd the Sttue of Lierty. 36. MODELING WITH MTHEMTIS The Lening Tower of Pis in Itly hs height of 183 feet nd is 4 off verticl. Find the horizontl distnce d tht the top of the tower is off verticl. d 7. m = 7, m = 44, = m = 98, m = 37, = m = 65, = 1, = ft 30. m = 90, = 15, c = m = 40, = 7, c = = 34, = 19, c = MODELING WITH MTHEMTIS You nd your friend re stnding on the seline of sketll court. You ounce sketll to your friend, s shown in the digrm. Wht is the distnce etween you nd your friend? (See Emple 8.) 37. MKING N RGUMENT Your friend sys tht the Lw of Sines cn e used to find JK. Your cousin sys tht the Lw of osines cn e used to find JK. Who is correct? Eplin your resoning. K 17 7 ft 6 ft 110 J 0 48 L 34. MODELING WITH MTHEMTIS zip line is constructed cross vlley, s shown in the digrm. Wht is the width w of the vlley? 38. RESONING Use XYZ. Z 17 5 ft 10 w 84 ft X 64. n you use the Lw of Sines to solve XYZ? Eplin your resoning. Y. n you use nother method to solve XYZ? Eplin your resoning. 514 hpter 9 Right Tringles nd Trigonometry

56 39. MKING N RGUMENT Your friend clcultes the re of the tringle using the formul = 1 qr sin S nd sys tht the re is pproimtely 08.6 squre units. Is your friend correct? Eplin your resoning. Q 17 S MODELING WITH MTHEMTIS You re fertilizing tringulr grden. One side of the grden is 6 feet long, nd nother side is 54 feet long. The ngle opposite the 6-foot side is Drw digrm to represent this sitution.. Use the Lw of Sines to solve the tringle from prt (). c. One g of fertilizer covers n re of 00 squre feet. How mny gs of fertilizer will you need to cover the entire grden? 41. MODELING WITH MTHEMTIS golfer hits drive 60 yrds on hole tht is 400 yrds long. The shot is 15 off trget. 15 Not drwn to scle 60 yd 400 yd u R 10 yd. Wht is the distnce from the golfer s ll to the hole?. ssume the golfer is le to hit the ll precisely the distnce found in prt (). Wht is the mimum ngle θ (thet) y which the ll cn e off trget in order to lnd no more thn 10 yrds from the hole? 4. OMPRING METHODS uilding is constructed on top of cliff tht is 300 meters high. person stnding on level ground elow the cliff oserves tht the ngle of elevtion to the top of the uilding is 7 nd the ngle of elevtion to the top of the cliff is 63.. How fr wy is the person from the se of the cliff?. Descrie two different methods you cn use to find the height of the uilding. Use one of these methods to find the uilding s height. 43. MTHEMTIL ONNETIONS Find the vlues of nd y. 3 5y 18 3y +.7 D 49 E F 44. HOW DO YOU SEE IT? Would you use the Lw of Sines or the Lw of osines to solve the tringle? in in REWRITING FORMUL Simplify the Lw of osines for when the given ngle is right ngle. 46. THOUGHT PROVOKING onsider ny tringle with side lengths of,, nd c. lculte the vlue of s, which is hlf the perimeter of the tringle. Wht mesurement of the tringle is represented y s(s )(s )(s c)? 47. NLYZING RELTIONSHIPS The miguous cse of the Lw of Sines occurs when you re given the mesure of one cute ngle, the length of one djcent side, nd the length of the side opposite tht ngle, which is less thn the length of the djcent side. This results in two possile tringles. Using the given informtion, find two possile solutions for. Drw digrm for ech tringle. (Hint: The inverse sine function gives only cute ngle mesures, so consider the cute ngle nd its supplement for.). m = 40, = 13, = 16. m = 1, = 17, = STRT RESONING Use the Lw of osines to show tht the mesure of ech ngle of n equilterl tringle is 60. Eplin your resoning. 49. RITIL THINKING n irplne flies 55 est of north from ity to ity, distnce of 470 miles. nother irplne flies 7 north of est from ity to ity, distnce of 890 miles. Wht is the distnce etween ities nd? Section 9.7 Lw of Sines nd Lw of osines 515

57 50. REWRITING FORMUL Follow the steps to derive the formul for the re of tringle, re = 1 sin. c. Drw the ltitude from verte to. Lel the ltitude s h. Write formul for the re of the tringle using h.. Write n eqution for sin. c. Use the results of prts () nd () to write formul for the re of tringle tht does not include h. 51. PROVING THEOREM Follow the steps to use the formul for the re of tringle to prove the Lw of Sines (Theorem 9.9).. Use the derivtion in Eercise 50 to eplin how to derive the three relted formuls for the re of tringle. re = 1 c sin, re = 1 c sin, re = 1 sin. Why cn you use the formuls in prt () to write the following sttement? 1 c sin = 1 c sin = 1 sin c. Show how to rewrite the sttement in prt () to prove the Lw of Sines. Justify ech step. 5. PROVING THEOREM Use the given informtion to complete the two-column proof of the Lw of osines (Theorem 9.10). Given D is n ltitude of. Prove = + c c cos c h STTEMENTS 1. D is n ltitude of. RESONS 1. Given D. D nd D re right tringles.. 3. = ( ) + h Epnd inomil h = c Sustitution Property of Equlity 7. cos = c = c cos = + c c cos 9. Mintining Mthemticl Proficiency Find the rdius nd dimeter of the circle. (Skills Review Hndook) Reviewing wht you lerned in previous grdes nd lessons ft ft 50 in. 10 in. 516 hpter 9 Right Tringles nd Trigonometry

58 Wht Did You Lern? ore Voculry trigonometric rtio, p. 488 cosine, p. 494 inverse cosine, p. 50 tngent, p. 488 ngle of depression, p. 497 solve right tringle, p. 503 ngle of elevtion, p. 490 inverse tngent, p. 50 Lw of Sines, p. 509 sine, p. 494 inverse sine, p. 50 Lw of osines, p. 511 ore oncepts Section 9.4 Tngent Rtio, p. 488 Section 9.5 Sine nd osine Rtios, p. 494 Sine nd osine of omplementry ngles, p. 494 Section 9.6 Inverse Trigonometric Rtios, p. 50 Solving Right Tringle, p. 503 Section 9.7 re of Tringle, p. 508 Theorem 9.9 Lw of Sines, p. 509 Theorem 9.10 Lw of osines, p. 511 Mthemticl Prctices 1. In Eercise 1 on pge 49, your rother clims tht you could determine how fr the overhng should etend y dividing 8 y tn 70. Justify his conclusion nd eplin why it works.. In Eercise 9 on pge 499, eplin the flw in the rgument tht the kite is 18.4 feet high. 3. In Eercise 31 on pge 506, for wht vlues does the inverse sine mke sense? Performnce Tsk Trithlon There is ig trithlon in town, nd you re trying to tke pictures of your friends t multiple loctions during the event. How fr would you need to wlk to move etween the photogrphy loctions? To eplore the nswers to this question nd more, go to igidesmth.com. 517

59 9 hpter Review 9.1 The Pythgoren Theorem (pp ) Dynmic Solutions ville t igidesmth.com Find the vlue of. Then tell whether the side lengths form Pythgoren triple c = + Pythgoren Theorem (Theorem 9.1) = = = 65 = 5 Sustitute. Multiply. dd. Find the positive squre root. The vlue of is 5. ecuse the side lengths 15, 0, nd 5 re integers tht stisfy the eqution c = +, they form Pythgoren triple. Find the vlue of. Then tell whether the side lengths form Pythgoren triple Verify tht the segment lengths form tringle. Is the tringle cute, right, or otuse? 4. 6, 8, nd ,, nd , 18, nd Specil Right Tringles (pp ) Find the vlue of. Write your nswer in simplest form. y the Tringle Sum Theorem (Theorem 5.1), the mesure of the third ngle must e 45, so the tringle is tringle. hypotenuse = leg Tringle Theorem (Theorem 9.4) = 10 = 10 Sustitute. Simplify The vlue of is 10. Find the vlue of. Write your nswer in simplest form hpter 9 Right Tringles nd Trigonometry

60 9.3 Similr Right Tringles (pp ) Identify the similr tringles. Then find the vlue of. 4 D Sketch the three similr right tringles so tht the corresponding ngles nd sides hve the sme orienttion. D D D D y the Geometric Men (ltitude) Theorem (Theorem 9.7), you know tht 4 is the geometric men of nd. 4 = Geometric Men (ltitude) Theorem 16 = Squre 4. 8 = Divide ech side y. The vlue of is 8. Identify the similr tringles. Then find the vlue of. 10. F 11. J K 9 6 E 6 H G M 4 L 1. R 3 S T Q P S 16 V U Find the geometric men of the two numers nd nd nd 4 hpter 9 hpter Review 519

61 9.4 The Tngent Rtio (pp ) Find tn M nd tn N. Write ech nswer s frction nd s deciml rounded to four plces. N opp. M tn M = dj. to M = LN LM = 6 8 = 3 4 = opp. N tn N = dj. to N = LM LN = 8 6 = M 8 L Find the tngents of the cute ngles in the right tringle. Write ech nswer s frction nd s deciml rounded to four deciml plces. 17. J 61 L K 18. M P 37 N Find the vlue of. Round your nswer to the nerest tenth The ngle etween the ottom of fence nd the top of tree is 75. The tree is 4 feet from the fence. How tll is the tree? Round your nswer to the nerest foot ft 9.5 The Sine nd osine Rtios (pp ) Find sin, sin, cos, nd cos. Write ech nswer s frction nd s deciml rounded to four plces. opp. sin = = hyp. = = opp. sin = = hyp. = = dj. to cos = = hyp. = = dj. to cos = = hyp. = = hpter 9 Right Tringles nd Trigonometry

62 Find sin X, sin Z, cos X, nd cos Z. Write ech nswer s frction nd s deciml rounded to four deciml plces. 4. Z 5. X 10 Y 6. Y 3 Y 5 4 X Z X Z Find the vlue of ech vrile using sine nd cosine. Round your nswers to the nerest tenth. 7. t 34 3 s 8. s 5 r w v Write sin 7 in terms of cosine. 31. Write cos 9 in terms of sine. 9.6 Solving Right Tringles (pp ) Solve the right tringle. Round deciml nswers to the nerest tenth. Step 1 Use the Pythgoren Theorem (Theorem 9.1) to find the length of the hypotenuse. c = + Pythgoren Theorem c = c = 505 c = 505 c.5 Sustitute. Simplify. Find the positive squre root. Use clcultor. c 19 1 Step Find m. Step 3 m = tn Use clcultor. 19 Find m. ecuse nd re complements, you cn write m = 90 m = In, c.5, m 3.3, nd m Let Q e n cute ngle. Use clcultor to pproimte the mesure of Q to the nerest tenth of degree. 3. cos Q = sin Q = tn Q = 0.04 Solve the right tringle. Round deciml nswers to the nerest tenth N 6 M Z L X Y hpter 9 hpter Review 51

63 9.7 Lw of Sines nd Lw of osines (pp ) Solve the tringle. Round deciml nswers to the nerest tenth.. y the Tringle Sum Theorem (Theorem 5.1), m = = y the Lw of Sines, you cn write 40 sin 40 = sin 65 = 0 sin sin 40 = 0 sin 75 0 sin 40 = sin 75 Write two equtions, ech with one vrile. Solve for ech vrile. = sin 65 = 0 sin 75 0 sin 65 sin Use clcultor In, m = 65, 13.3, nd First, find the ngle opposite the longest side,. Use the Lw of osines to find m. 19 = (11)(17) cos Lw of osines = cos (11)(17) Solve for cos. m 8.5 Use clcultor. Now, use the Lw of Sines to find m. sin = sin c sin sin 8.5 = sin 8.5 sin = 19 m 35.0 Lw of Sines Sustitute. Multiply ech side y 11. Use clcultor. y the Tringle Sum Theorem (Theorem 5.1), m = 6.5. In, m 35.0, m 6.5, nd m 8.5. Find the re of with the given side lengths nd included ngle. 38. m = 14, = 9, c = m = 68, = 13, c = m = 79, = 5, = 17 Solve. Round deciml nswers to the nerest tenth. 41. m = 11, = 9, = 4 4. m = 8, m = 64, c = m = 48, = 0, c = m = 5, = 8, c = m = 10, m = 43, = = 10, = 3, c = 1 5 hpter 9 Right Tringles nd Trigonometry

64 9 hpter Test Find the vlue of ech vrile. Round your nswers to the nerest tenth. 1. t 18 5 s. y 6 3. j k Verify tht the segment lengths form tringle. Is the tringle cute, right, or otuse? 4. 16, 30, nd , 67, nd , 5, nd 5.5 Solve. Round deciml nswers to the nerest tenth m = 103, = 1, c = m = 6, m = 35, = = 38, = 31, c = Write cos 53 in terms of sine. Find the vlue of ech vrile. Write your nswers in simplest form q 45 r 15. c e d 16. h 30 8 f 17. In QRS, m R = 57, q = 9, nd s = 5. Find the re of QRS. 18. You re given the mesures of oth cute ngles of right tringle. n you determine the side lengths? Eplin. 19. You re t prde looking up t lrge lloon floting directly ove the street. You re 60 feet from point on the street directly eneth the lloon. To see the top of the lloon, you look up t n ngle of 53. To see the ottom of the lloon, you look up t n ngle of 9. Estimte the height h of the lloon. 53 h viewing ngle 9 60 ft 0. You wnt to tke picture of sttue on Ester Islnd, clled moi. The moi is out 13 feet tll. Your cmer is on tripod tht is 5 feet tll. The verticl viewing ngle of your cmer is set t 90. How fr from the moi should you stnd so tht the entire height of the moi is perfectly frmed in the photo? hpter 9 hpter Test 53

65 9 umultive ssessment 1. The size of lptop screen is mesured y the length of its digonl. You wnt to purchse lptop with the lrgest screen possile. Which lptop should you uy? 1 in. 0 in. 9 in in. 1 in. D 8 in in. 6 in.. In PQR nd SQT, S is etween P nd Q, T is etween R nd Q, nd QS SP = QT TR. Wht must e true out ST nd PR? Select ll tht pply. ST PR ST PR ST = PR ST = 1 PR 3. In the digrm, JKL QRS. hoose the symol tht mkes ech sttement true. J Q R 4 K 8 L S sin J sin Q sin L cos J cos L tn Q cos S cos J cos J sin S tn J tn Q tn L tn Q tn S cos Q sin Q cos L < = > 4. surveyor mkes the mesurements shown. Wht is the width of the river? ft hpter 9 Right Tringles nd Trigonometry

66 5. rete s mny true equtions s possile. X Y 3 3 = Z sin X cos X tn X XY YZ XZ XZ sin Z cos Z tn Z XY YZ YZ XY 6. Prove tht qudrilterl DEFG is kite. Given HE HG, EG DF Prove FE FG, DE DG D H E F G 7. Wht re the coordintes of the vertices of the imge of QRS fter the composition of trnsformtions shown? Trnsltion: (, y) ( +, y + 3) Rottion: 180 out the origin y Q (1, ), R (5, 4), S (4, 1) R Q ( 1, ), R ( 5, 4), S ( 4, 1) Q 4 Q (3, ), R ( 1, 4), S (0, 1) D Q (, 1), R ( 4, 5), S (1, 4) 4 S 8. The Red Pyrmid in Egypt hs squre se. Ech side of the se mesures 7 feet. The height of the pyrmid is 343 feet.. Use the side length of the se, the height of the pyrmid, nd the Pythgoren Theorem to find the slnt height,, of the pyrmid. 1. Find. c. Nme three possile wys of finding m 1. Then, find m 1. D hpter 9 umultive ssessment 55