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# College Trigonometry MATH 124

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This 38 page Class Notes was uploaded by Lisa Wisoky on Monday October 12, 2015. The Class Notes belongs to MATH 124 at Fayetteville State University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/221585/math-124-fayetteville-state-university in Mathematics (M) at Fayetteville State University.

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Date Created: 10/12/15

Chapter 3 Analytic Trigonornetrv 3 1 The inverse sine cosine and tangent functions 1 Review Inverse function 1 f 1fx a for every 3 in the dornain of f and ff 1x a for every 3 in the dornain of 1 2 Dornain of f range of 1H and range of f dornain of f 1 3 The graph of f and the graph of f 1 are svrnrnetric with respect to the 1ine y a 4 If a function y fc has an inverse function the equation of the inverse function is a f y The so1ution of this equation is y f 1x 2 The inverse sine function x s m um 3 5 mus 3 s um um i w n V i If We restrict the dornain of y sinr to e1 2 a the restrict function y 7r 7r W lt lt 7 sin cc 2 g a g 2 will have aninverse function We call it the ofoc We denote it by y sin 1 a rneans a sing Whereil r lmdig y g 3 Graph y sin 1 a Dornain of y sin 1 a is and range is 4 Example Find the exact value of sin 17 5 Remark sin 1sin L where sinsin 1 L where 6 Example Find the exact value of 1 sin 1sing 2 sinsin 17 7 The inverse cosine function ax m me W m mg n reps m m m mm W t r v If We restrict the domain of y nose to 07r the restrict function ycosa 0Sxlt7r will have an inverse function We call it the of a We denote it by ycos acmeansascosy wheree1gxg1and0gyg7r 8 Graph y cosquot a Domain of y 00539102 is and range is Ehaannple Fmd the exact value at cosquotr 10 Remark cosquotcos e where 0 g e g coscosquot e e where 71 e 1 11 Example Fmd the exut values of 1 cosquotcos 2 coscosquot70 m 12 The Inverse tangent tunctmn NH we nestnct the domain of 3 tane to 77 g the restrict tunctmn t tan at 0 mu have an Inverse tunctmn We call 1t the out We denote 1t by y tanquot 0 means tan t lt y lt quotE where roolt at lt co and 13 Graph y tan 1x Domain of y tan 1 x is and range is 14 Example Find the exact Value of tan 17 15 Remark tan 1tanx x Where 7 lt x lt tantan 1x x where 700 lt x lt 00 16 Example Find the exact Values of 1 tan 1ltanl 2 tantan 1 32 The inverse trigonometric functions Continued 1 Example Find the exact value of sin 1sin 2 Example Find the exact value of sintan 1 3 Example Find the exact value of cossin 17 4 Example Find the exact value of tancos 17 5 The remaining inverse trigonometric functions y sec 1 z means z secy7 where 21 and 0 S y M 11 7r 7 5 y csc 1 z means z cscy7 where 2 1 and 7 g y 31 0 y cot ls means z coty where foo lt z lt 00 and 0 lty lt 7139 S I 2 6 Example Find the exact value of csc 1 2 33 Trigonornetric identities 1 Basis trigonometric identities D Quotient identities Reciprocal identities Pythagorean identities Even odd identities Example Simplify the following expressions cot9 csc 9 1 cos 9 1sin 9 2 1sin9 cot 97cos9 i 9 c059 tan9in 9tan9 3 Example Establish the following identities 1 0300 tam sect 2sin276 003276 1 3 W COS0 isin 1t 9 7 4 Heft itan 5 6 7 sin 9 1cos 9 1H0 2 csc 9 sin9 tan 9cot 9 1 sec 9csc 9 lisin9 cos9 cos9 1sin9 i V1111 n sinta 8 34 Sum and difference formulas 1 Sum and difference formulas for cosmes cosa B cos Oz cos 7 sin 04 sm cosa 7 B cosa cos smasm 2 Example Find the exact values of cos 750 and cos 3 cos 7 l9 sm0 sin 7 l9 cos 6 4 Sum and difference formulas for Sines sina B sin a cos cos a sin sina 7 B sina cos B 7 cos a sin 5 Example Find the exact values of sin 177239 and sin800 cos 200 7 cos 800 sin 200 6 Example If it is known that sina g glt04lt7T7andlthatsin 77 7Tlt lt7 nd the exact value of 1 cosa 2 cos 3 cosa B 4 sma B 7 Example Establish the identity cot acot 1 8 Sum and difference formulas for tangents tana tanoztan i tanaitan tana i 1tanuztan 9 Example Prove the identity tan6 7T tan0 10 Example Prove the identity tan6 7cot 0 11 Example Find the exact value of sincos 1 sin l 12 Example Write sinsin 1u cos 1 1 as an algebraic expression containing u and 1 that is7 without any trigonometric functions 35 Double angle and half angle formulas 1 Double angle formulas sm26 2 sm 6cos 6 cos26 cos2 6 7 sin2 6 cos26 1 7 2 sm26 cos26 2 cos2 6 71 D Example If sm6 27 lt 6 lt 7T nd the exact values of sm26 and cos26 3 Example 1 Develop a formula for tan26 in terms of tan 6 2 Develop a formula for sm36 in terms of sm6 and cos 6 4 Other variations of the double angle formulas licos29 1cos29 2 i licos29 2 7 2 7 tan 6 1cos29 sin2 6 cos2 6 5 Example Write an equivalent expression for cos4t9 that does not involve any powers of sine or cosine greater that 1 6 Half angle formulas g licosa g 1cosoz g licosa sinzii 2 coszii 2 tanzii Hm where the and 7 sign is determined by the quadrant of the angle 7 Example Use half angle formulas nd the exact values of cos 150 and sin7150 8 Example If cosa 7 7139 lt a lt 37739 nd the exact value of 1 sin 2 003 3 tan 9 Half angle for tan 0 130504 sine tan 2 i sine i 1cosa39 36 Product to sum and sum to product formulas 1 Product to sum formulas sin 04 sin 6 cosa 7 6 7 cosa 6 cosa cos6 cosoz 7 6 cosa 6 sinacos6 sinoz 6 sina 7 6 1 sin6t9 sin46 2 Example Express each of the following products as a sum containing only sines or cosines 2 cos36 cos 6 3 sin36 cos56 3 Sum to product formulas sin 04 7 sin6 2 sin a 7 TcosT7 sina7sin6 2sinLg cosLJg cosozcos62cos0 2i cos0 7 cosoz7cos672sinL sinL 4 Example Express each sum or difference as a product of sines and0r cosines 1 sin56 7 sin36 2 cos36 cos2t9 37 Trigonometric equations 1 1 Example Determine whether 6 E is a solution of the equation sin0 ls t9 g a solution 2 Example Solve the equation cost Give a general formula for all solutions List six solutions 3 Example Solve the equation 2 sin6 07 0 S 6 lt 27139 4 Example Solve the equation sin26 0 S 6 lt 27139 1 E7 5 Example Solve the equation tan6 7 17 0 S 6 lt 27139 38 Trigonometric equations H 1 Example Solve the equation 2 sin2 6 7 33in6 1 07 0 S 6 lt 27139 2 Example Solve the equation 300s6 3 2 sin2 67 0 S 6 lt 27139 3 Example Solve the equation 00326 3 500367 0 S 6 lt 27139 4 Example Solve the equation COSZ 6 sin6 27 0 S 6 lt 27139 5 Example Solve the equation sin6cos 6 7 0 S 6 lt 27139 6 Example Solve the equation sin6 0036 17 0 S 6 lt 27139 REVIEW 2 MATH 124 COLLEGE TRIGONOMETRY 1 Find the exact value of each expression 1 sin 1 g 2 cos 1 73 3 cossin 1 4 tancos 1 2 Establish each identity 1 0800 cos6 cot6 2 cos 0tan 0 cot 6 csc 0 lisint9 cost i2se00 cos 9 lisint i 3 Find the exact value of each trigonometric function 77r 1 s1n E 2 cos 4 Find the exact value of each of the following under the given conditions a mm m b mm m o my m d tanlta m lsin0z 0lt0zltcos 7 lt lt0 4 1 2tana lt0zlt7rcos 0lt ltg 5 Find the exact value of each expression 1 sinsin 1 5 COS 1 71 4 71 5 2 costan g cos E 6 Use the information given about the angle 6 0 S 0 S 27F to nd the exact value of a sin20 b cos26 0 sin d cosg 1 sin6 0lt0ltg 2tau116 77rlt0lt377r 7 Find the exact value of each expression 1 sin2 sin 1 2 cos2 sin 1 8 Establish the identity tan36 9 Solve each equation on the interval 0 S 0 lt 2m 1 2sin03 2 2 cos26 NIH 3 200s26cos6 0 4 2sin26 sin6 10

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