A Trigonometric Fnctions (pp 8 ) Rtios of the sides of right tringle re sed to define the si trigonometric fnctions These trigonometric fnctions, in trn, re sed to help find nknown side lengths nd ngle mesres of tringles A Trigonometric Fnctions Voclr The si trigonometric fnctions nd their revitions re: sine, or sin cosine, or cos tngent, or tn cotngent, or cot secnt, or sec cosecnt, or csc Evlte Trigonometric Fnctions In right tringle with cte ngle, the si trigonometric fnctions re defined s follows: sin 5 opposite side djcent side opposite side } cos 5 } tn 5 } hpotense hpotense djcent side csc 5 } hpotense sec 5 hpotense djcent side } cot 5 } opposite side djcent side opposite side Evlte the si trigonometric fnctions of the ngle 7 hpotense 4 From the Pthgoren Theorem, the hpotense hs length Ï } 7 4 5 Ï } 65 5 5 sin 5 opp } hp 5 4 } 5 cos 5 dj } hp 5 7 } 5 tn 5 opp } dj 5 4 } 7 csc 5 hp } opp 5 5 } 4 sec 5 hp } dj 5 5 } 7 cot 5 dj } opp 5 7 } 4 Evlte the si trigonometric fnctions of the ngle 9 4 5 6 Copright McDogl Littell, division of Hoghton Mifflin Compn 8 Benchmrk 6 Chpters nd 4
Solve Right Tringle B Solve ABC A nd B re complementr ngles, so B 5 90 60 5 0 C c 60 5 4 A csc 5 } sin sec 5 } cos cot 5 } tn tn 60 5 opp } dj sec 60 5 hp } dj Write trigonometric eqtion tn 60 5 } 4 sec 60 5 c } 4 Sstitte 5 4(tn 60 ) c 5 4(sec 60 ) 5 4 } cos 60 Solve for the vrile 5 4Ï } c 5 4 } } Solve ABC sing the digrm nd given mesrements 5 4 5 8 Evlte trigonometric fnctions 4 B 5 45, c 5 Ï } 5 A 5 0, 5 6 B C c A A Trignometric Fnctions Drw Generl Angles Copright McDogl Littell, division of Hoghton Mifflin Compn Voclr The mesre of n ngle is positive if the rottion of the terminl side is conterclockwise If the rottion is clockwise, the ngle mesre is negtive Initil side The fied r of n ngle on coordinte plne Terminl side The r of n ngle tht is rotted ot the verte on coordinte plne Stndrd position The loction of n ngle whose verte is t the origin nd initil side lies on the positive -is 80 terminl side verte Drw n ngle with the given mesre in stndrd position 0 480 c 5 Since 0 is 0 more thn 80, the terminl side is 0 conterclockwise pst the negtive -is 0 0 90 70 initil side 0 60 Benchmrk 6 Chpters nd 4 9
Since 480 is 0 more thn 60, the terminl side mkes one whole revoltion conterclockwise pls 0 more 0 480 A Trigonometric Fnctions Voclr To convert degrees to rdins, mltipl degrees rdins } 80 To convert rdins to degrees, mltipl rdins 80 } rdins c Since 5 is negtive, the terminl side is 5 clockwise from the positive -is Drw n ngle with the given mesre in stndrd position 6 00 7 5 8 600 4 Convert Between Degrees nd Rdins 5 Rdin The mesre of n ngle in stndrd position whose terminl side intercepts n rc of length r Convert () 5 to rdins nd () to degrees rdins 5 5 5 } 80 5 } 5 4 rdins, or simpl } 5 4 } 5 } 80 rdins } 5 0 rdins Convert the degree mesre to rdins or the rdin mesre to degrees 7 9 50 0 } } 6 9 r r Copright McDogl Littell, division of Hoghton Mifflin Compn 0 Benchmrk 6 Chpters nd 4
For n ngle in stndrd position whose terminl side intersects circle with rdis r, the trigonometric fnctions re defined s follows: sin 5 } r cos 5 } r tn 5 }, 0 Voclr 5 Evlte Trigonometric Fnctions Given Point The point (, 4) is point on the terminl side of n ngle in stndrd position Evlte the si trigonometric fnctions of B the Pthgoren Theorem, r 5 Ï } 5 Ï } () 4 5 Ï } 5 5 5 For 5, 5 4, nd r 5 5, (, 4) r sin 5 } r 5 4 } 5 cos 5 } r 5 } 5 tn 5 } 5 4 } csc 5 } sin 5 5 } 4 sec 5 } cos 5 5 } cot 5 } tn 5 } 4 Evlte the si trigonometric fnctions with the given point on the terminl side of ngle (6, 8) (8, 5) 4 (, 5) 6 Use Reference Angles Reference ngle The cte ngle 9 formed the terminl side of n ngle in the stndrd position nd the -is A Trignometric Fnctions Copright McDogl Littell, division of Hoghton Mifflin Compn Signs of trigonometric fnctions in ech qdrnt re s follows: Qdrnt I sin 5 cos 5 tn 5 Qdrnt II sin 5 cos 5 tn 5 Qdrnt III sin 5 cos 5 tn 5 Qdrnt IV sin 5 cos 5 tn 5 9 9 Use reference ngles to evlte () tn (5 ) nd () cos 8 The ngle 5 is coterminl with 5 The reference ngle is 80 5 5 45 The tngent fnction is negtive in Qdrnt II, so tn (5 ) 5tn 45 5 9545 55 9 Benchmrk 6 Chpters nd 4
The ngle 8 } is coterminl with } The reference ngle is } 5 } 95 The cosine fnction is negtive in Qdrnt II, so cos 8 } 5cos } = } 5 8 Evlte the trigonometric fnction withot sing clcltor A Trigonometric Fnctions For sin 5, the inverse sine is sin 5, 90 90 For cos 5, the inverse cosine is cos 5, 0 80 For tn 5, the inverse tngent is tn 5, 90 90 5 cos 00 6 csc 0 7 tn 7 Evlte Inverse Trigonometric Fnctions, Solve Trigonometric Eqtion Evlte () sin 05 nd () tn When 90 90, the ngle whose sine is 05 is: 5 sin 05 5 0 When 90 90, the ngle whose tngent is Ï } is: 5 tn Ï } 5 60 7 } 6 A 7-meter rmp hs horizontl length of 5 meters Wht is the ngle of the rmp? Drw tringle tht represents the rmp Write trigonometric eqtion tht involves the rmp s length nd horizontl length cos 5 } 5 7 Use clcltor to find the mesre of 5 5 cos } 7 < 8 Evlte the epression 7 m 5 m 8 tn 9 cos 05 0 sin Ï } A cle wire is ttched to the top of 0-foot pole 6 feet from the se of the pole Wht is the ngle the wire mkes with the grond? Copright McDogl Littell, division of Hoghton Mifflin Compn Benchmrk 6 Chpters nd 4
Qiz Evlte the si trigonometric fnctions of the ngle 4 6 5 5 Solve ABC sing the digrm nd given mesrements A 5 45, 5 4 B 5 60, c 5 5 Convert the degree mesre to rdins or the rdin mesre to degrees 5 75 6 } 7 50 Evlte the si trigonometric fnctions with the given point on the terminl side of ngle 8 (0, 5) 9 (4, 7) 0 (, ) B C c A A Trignometric Fnctions Copright McDogl Littell, division of Hoghton Mifflin Compn Evlte withot sing clcltor tn (0 ) cos 5 } cos Ï} } 4 sin Ï} } 5 An irplne egins its descent for lnding t n ltitde of 9,000 feet At this time, the irplne is 50 miles from the rnw At wht ngle does the irplne descend? Benchmrk 6 Chpters nd 4
B Lw of Sines nd Lw of Cosines (pp 4 6) The si trigonometric rtios cn e sed to solve right tringles When tringle contins no right ngles, formls relting to sine nd cosine cn e sed to solve the tringle B Lw of Sines nd Cosines Voclr For nabc with opposite sides,, nd c, the lw of sines is: sin A } 5 } sin B 5 } sin C c Use the Lw of Sines Lw of sines A method for solving tringle when two ngles nd side re known (AAS or ASA cses) or when the lengths of two sides nd n ngle opposite one of those sides re known (SSA cse) Solve ABC with A 5, C 5 7, nd c 5 7 First find the third ngle: B 5 80 7 5 4 B the lw of sines, sin sin 4 sin 7 } 5 } 5 } 7 sin sin 7 } 5 } Write two eqtions with one vrile 7 sin 4 sin 7 } 5 } 7 7 sin 5 } sin 7 Solve for ech vrile 7 sin 4 5 } sin 7 < 4 Use clcltor < 0 Solve ABC A 5 45, B 5 6, c 5 0 C 5 05, B 5 0, 5 6 Voclr In the digrm t right, h 5 sin A Emine SSA Tringles SSA cse When the lengths of two sides nd n ngle opposite one of those sides re known, reslting in either no tringle, one tringle, or two different tringles A A A is otse # No tringle One tringle A A h h h h,, Two tringles A is cte A h h 5 One tringle h One tringle Determine the nmer of tringles tht cn e formed Solve the tringle if onl one tringle cn e formed A 5 4, 5, 5 8 A 5 0, 5 6, 5 0 Copright McDogl Littell, division of Hoghton Mifflin Compn c A 5 55, 5, 5 4 4 Benchmrk 6 Chpters nd 4
Since A is otse nd the side opposite A is longer thn the given djcent side, onl one tringle cn e formed Use the lw of sines to solve the tringle A 4 8 B C Copright McDogl Littell, division of Hoghton Mifflin Compn Voclr For ABC with opposite sides,, nd c, thelw of cosines is: 5 c c cos A 5 c c cos B c 5 cos C sin 4 } 5 } sin B Lw of sines 8 sin B 5 B < 07 8 sin 4 } < 050 Mltipl ech side 8 Use inverse sine fnction C < 80 4 07 5 5 sin 4 sin 5 } 5 } c Lw of sines c 5 sin 5 } sin 4 < 67 Cross mltipl Since A is otse nd the side opposite A is shorter thn the given djcent side, it is not possile to drw the indicted tringle No tringle eists with these given sides nd ngle c Since sin A 5 4 sin 55 < 5, nd 5,, 4 (h,, ), two tringles cn e formed Determine the nmer of tringles tht cn e formed 4 C 4 55 55 A B A Tringle Tringle A 5 60, 5 6, 5 9 4 A 5 8, 5 0, 5 5 5 A 5 4, 5 8, 5 5 Find n Unknown Side with the Lw of Cosines Lw of cosines A method for solving tringle when two sides nd the inclded ngle re known (SAS cse) or when ll three sides re known (SSS cse) Find the nknown side in ABC when 5 6, c 5, nd A 5 75 5 c c cos A Lw of cosines 5 6 (6)() cos 75 Sstitte for, c, nd A < 006 < Ï } 006 < 7 Simplif Tke positive sqre root C Benchmrk 6 Chpters nd 4 5 B B Lw of Sines nd Cosines
Find the nknown side in ABC 6 5 7, c 5 5, nd B 5 08 7 5 4, 5 6, nd C 5 9 4 Find Unknown Angles with the Lw of Cosines Solve ABC with 5 9, 5, nd c 5 0 B Lw of Sines nd Cosines Find the ngle opposite the longest side sing the lw of cosines 5 c c cos B Lw of cosines 5 9 0 (9)(0) cos B Sstitte 9 0 }} (9)(0) 5 cos B Solve for cos B 0056 < cos B Simplif B < 78 Use inverse cosine Now se the lw of sines sin A } 5 } sin B Lw of sines sin A sin 78 } 9 5 } Sstitte sin A 5 9 sin 78 } < 079 Mltipl ech side 9 nd simplif A < 47 Use inverse sine The third ngle of the tringle is C < 80 47 78 5 547 8 Solve nabc with 5 6, 5 7, nd c 5 Qiz Determine the nmer of tringles tht cn e formed A 5 4, 5 6, 5 A 5 54, 5 4, 5 7 A 5 99, 5 8, 5 8 Solve ABC 4 B 5 50, A 5 8, 5 5 5 A 5 7, C 5 97, 5 9 6 5, c 5 8, A 5 00 7 5 4, 5 5, C 5 6 8 5 5, 5, c 5 4 9 5 0, 5 4, c 5 Copright McDogl Littell, division of Hoghton Mifflin Compn 6 Benchmrk 6 Chpters nd 4
C Grph Trigonometric Fnctions (pp 7 9) Yo lerned how to se sine, cosine, nd tngent fnctions to solve right tringles Here o will lern how to grph these fnctions on the coordinte plne Copright McDogl Littell, division of Hoghton Mifflin Compn Voclr Given nonzero rel nmers nd in fnctions 5 sin nd 5 cos, the mplitde of ech is nd the period of ech is } Given nonzero rel nmers nd in the fnction 5 tn, the period is } The verticl smptotes re odd mltiples of } There re ( ) no mimm or minimm vles, so there is no mplitde Grph Sine nd Cosine Fnctions Amplitde Hlf the difference etween fnction s mimm M nd minimm m Periodic fnction A fnction with repeting pttern Ccle The shortest repeting portion of grph Period The horizontl length of ech ccle Grph () 5 sin nd () 5 cos The mplitde is 5 nd the period is } 5 } 5 Intercepts: (0, 0); }, 0 5 (, 0); (, 0) Mimm: } 4, 5 }, Minimm: } 4, 5 }, The mplitde is 5 nd the period is } 5 } Intercepts: } 4 }, 0 5 } 6, 0 ; } 4 }, 0 5 }, 0 Mimms: (0, ); }, Minimm: } }, 5 }, Grph the fnction 4 4 5 5 cos 5 sin 5 sin 4 Grph Tngent Fnction Grph one period of the fnction 5 tn The period is } 5 } Intercept: (0, 0) Asmptotes: 5 } 5 } or } 4 ; 5 } 5 } or } 4 Hlfw points: } 8, nd } 8, 8 6 4 4 4 6 8 Benchmrk 6 Chpters nd 4 7 4 C Grph Trigonometrics
Voclr Grph one period of the fnction 4 5 4 tn 5 5 tn 6 5 tn } Grph Trnsltions nd Reflections Trnsltion of trigonometric fnction A horizontl shift h nits, verticl shift k nits, or comintion of oth horizontl shift nd verticl shift in the grph of fnction Reflection of trigonometric fnction A flip cross horizontl line eqidistnt from the mimm nd minimm points on fnction s grph Midline The horizontl line fnction is reflected cross C Grph Trigonometrics The grphs of 5 sin ( h) k nd 5 cos ( h) k re shifted horizontll h nits nd verticll k nits from their prent fnctions Grph 5 sin } ( ) Step : Identif the mplitde, period, horizontl shift, nd verticl shift Amplitde: 5 5 ; Period: } } 5 } 5 4 Horizontl shift: h 5; Verticl shift: k 5 0 Step : Drw the midline of the grph Since k 5 0, the midline is the -is Step : Find five ke points of 5 sin } ( ) On the midline 5 k: (0, 0) 5 (, 0); (, 0) 5 (, 0); (4, 0) 5 (, 0) Mimm: (, ) 5 (0, ) Minimm: (, ) 5 (, ) Step 4: Reflect the grph Since 0, the grph is reflected in the midline 5 0 So, (0, ) ecomes (0, ) nd (, ) ecomes (, ) Step 5: Drw the grph throgh the ke points Grph the fnction 7 5cos } ( ) 8 5 sin } 4 4 Copright McDogl Littell, division of Hoghton Mifflin Compn 8 Benchmrk 6 Chpters nd 4
Qiz Grph one period of the fnction 5 6 sin 5 cos 5 tn } 4 4 5 cos 4 5 5 4 sin } 6 5 tn 7 5 sin } 8 5 4 cos } 9 5tn } 4 ( ) 0 5 sin } ( ) C Grph Trigonometrics Copright McDogl Littell, division of Hoghton Mifflin Compn Benchmrk 6 Chpters nd 4 9
D Trigonometric Identities (pp 0 ) Fndmentl trigonometric identities tht cn e sed to simplif epressions, evlte fnctions, nd help solve eqtions Severl trigonometric identities re descried nd pplied elow D Trigonometric Identities Voclr Trigonometric cofnction identities inclde: sin } 5 cos cos } 5 sin sec } 5 csc csc } 5 sec tn } 5 cot cot } 5 tn Trigonometric Pthgoren identities inclde: sin cos 5 tn 5 sec cot 5 csc sin ( ) Þ sin sin sin ( ) Þ sin sin The sme is tre for the other trigonometric fnctions Use Fndmentl Trigonometric Identities Trigonometric identit An epression or eqtion involving trigonometric fnctions tht is tre for ll vles of the vrile cos Simplif the epression }} tn } cos } } tn 5 } sin tn Sstitte sin for cos } Simplif the epression 5 } sin Sstitte sin } cos 5 cos Simplif sin } cos for tn cot sec } csc csc sin csc cot (sec ) Use Sm nd Difference Formls Trigonometric fnctions relting to the sm nd difference of two ngles re s follows: sin ( ) 5 sin cos cos sin sin ( ) 5 sin cos cos sin cos ( ) 5 cos cos sin sin cos ( ) 5 cos cos sin sin tn ( ) 5 tn tn }} tn tn tn ( ) 5 tn tn }} tn tn Copright McDogl Littell, division of Hoghton Mifflin Compn 0 Benchmrk 6 Chpters nd 4
Find the ect vle of tn } tn } 5 tn } 4 } 6 Sstitte 4 6 for tn } 4 tn } 6 5 }} tn } 4 tn } Difference forml for tngent 6 Copright McDogl Littell, division of Hoghton Mifflin Compn The sine of n ngle in Qdrnt II is positive The cosine of n ngle in Qdrnt II is negtive 5 Ï} } } Evlte Ï} } 5 Ï } Simplif Find sin ( ) given tht cos 5 } 5 with } nd sin 5 8 } 7 with 0 } Using Pthgoren identit nd qdrnt signs gives sin 5 4 } 5 nd cos 5 5 } 7 sin ( ) 5 sin cos cos sin 5 4 } 5 5 } 7 } 5 8 } 7 Sstitte 5 84 } 85 Simplif Find the ect vle of the epression Difference forml for sine 4 cos 75 5 tn 05 6 sin 7 Find cos ( ) given tht sin 5 } with } nd sin 5 4 } 5 with } Use Dole-Angle nd Hlf-Angle Formls Dole-ngle formls: sin 5 sin cos Hlf-ngle formls: sin } cos } 56 Ï } cos 5 cos sin } cos cos } 56 Ï } 5 cos tn } 5 } cos sin } D Trigonometric Identities 5 sin 5 sin } cos tn 5 tn } tn Benchmrk 6 Chpters nd 4
Becse 5 is in Qdrnt I, the vle of the sine is positive Find the ect vle of sin 5 } sin 5 5 sin } } cos 0 (0 ) 5 Ï } 5 Ï Ï} } } 5 Ï} Ï } } D Trigonometric Identities Give sin 5 } 5 with }, find sin nd sin } Using Pthgoren identit nd qdrnt signs gives cos 5 4 } 5 sin 5 sin cos 5 } 5 4 } 5 5 4 sin } } 5 } cos } } 5 Ï } 5 Ï 4 } 5 9}0 } 5 Ï 5 Ï} 0 } 0 Given cos 5 } 5 with p 8 cos 9 tn Qiz Simplif the epression cos tn cos cos } p, find ech vle } 0 cos } cos F cot } G sec Find the ect vle of the epression 4 sin 05 5 cos } 6 tn 55 7 Find cos ( ) given tht sin 5 4 } 5 with } nd cos 5 4 } 5 with } Simplif the epression 8 sin ( ) 9 tn ( ) 0 cos } Copright McDogl Littell, division of Hoghton Mifflin Compn Benchmrk 6 Chpters nd 4
E Solve Trigonometric Eqtions (pp 5) When trigonometric fnction is tre for ll vles of the vrile, the fnction is n identit When fnction is tre onl for some vles of the vrile, the fnction is n eqtion The methods for finding the tre vles of the vrile in trigonometric eqtion re presented elow Solve Trigonometric Eqtion Solve sin 0 First isolte sin on one side of the eqtion sin 5 0 Write originl eqtion sin 5 Add to ech side sin 5 } Divide ech side One soltion of sin 5 } over the intervl 0 is 5 sin } 5 } 6 The other soltion in this intervl is 5 } 6 5 } 5 6 Since 5 sin is periodic, there re infinitel mn soltions Using the two soltions ove, the generl soltion is written s follows: E Solving Trigonometrics 5 } 6 n or 5 5 } 6 n, for n integer n Copright McDogl Littell, division of Hoghton Mifflin Compn It is helpfl to memorize the trigonometric vles for specil ngles 0, 0, 45, 60, 90, nd 80 Solve 8 cos in the intervl 0 8 cos 5 Write originl eqtion 8 cos 5 Strct from ech side cos 5 } Divide ech side 8 4 cos 56 } Tke sqre roots of ech side One soltion of cos 56 } is 5 cos } 5 } Another soltion is 5 cos } 5 } Over the intervl 0, the soltions re: 5 } 5 } 5 } 5 } 5 } 4 5 } 5 } 5 Find the generl soltion of the eqtion tn 4 5 4 sin 5 5 Solve the eqtion in the intervl 0 cos 5 4 tn 5 8 Benchmrk 6 Chpters nd 4
Solve n Eqtion with Etrneos Soltions Solve sin cos in the intervl 0 sin 5 cos Write originl eqtion ( sin ) 5 (cos ) Sqre oth sides sin sin 5 cos Mltipl sin sin 5 sin Pthgoren identit sin sin 5 0 Qdrtic form E Solving Trigonometrics It is good prctice to lws check soltions in trigonometric eqtions since it is possile for these eqtions to contin etrneos soltion Voclr In the sinsoid = sin ( h) + k, is the mplitde, is sed to find the period, h is the horizontl shift, nd k is the verticl shift sin (sin ) 5 0 Fctor ot sin sin 5 0 or sin 5 0 Zero prodct propert sin 5 0 or sin 5 Solve for sin Over the intervl 0, sin 5 0 hs two soltions: 5 0 or 5 Over the intervl 0, sin 5 hs one soltion: 5 } Therefore, sin 5 cos hs three possile soltions: 5 0, }, nd Check Sstitte the possile soltions into the originl eqtion nd simplif 5 0: sin 0 5 cos 0 0 5 5 Soltion checks 5 } : sin } 5 cos } 5 0 05 0 Soltion checks 5 : sin 5 cos 0 5 Soltion is etrneos The onl soltions to sin 5 cos in the intervl 0 re 5 0 nd 5 } Solve the eqtion in the intervl 0 5 cos 5 sin cos 6 tn sin tn 5 0 Write Sinsoid Sinsoid A grph of sine or cosine fnction Write fnction for the sinsoid Step : Find the mimm vle M nd the minimm vle m From the grph, M 5 nd m 5 Step : Identif the verticl shift, k This vle is the men of M nd m So, k 5 4 4 4 4 Copright McDogl Littell, division of Hoghton Mifflin Compn 4 Benchmrk 6 Chpters nd 4
Step : Decide whether the grph models sine or cosine fnction Since the grph crosses the midline 5 on the -is, the grph is sine crve with no horizontl shift So, h 5 0 Step 4: Find the mplitde nd period The period } 5 } so 5 The mplitde 5 } M m 5 } () 5 The grph is not reflection, so 5 The fnction is 5 sin Write fnction for the sinsoid 7 Qiz 4 4 Find the generl soltion of the eqtion 8 4 5 6 4 sin 4 5 0 8 cos 5 7 tn 5 E Solving Trigonometrics Copright McDogl Littell, division of Hoghton Mifflin Compn Solve the eqtion in the intervl 0 4 6 sin 5 5 5 cos cot 5 sin 6 sin Ï } cos 5 Write fnction for the sinsoid 7 6 5 4 8 6 6 Benchmrk 6 Chpters nd 4 5