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CHAPTER 7: Trigonometric Identities, Inverse Functions, and Equations 7.1 Identities: Pythagorean and Sum and Difference 7.2 Identities: Cofunction, Double-Angle, and HalfAngle 7.3 Proving Trigonometric Identities 7.4 Inverses of the Trigonometric Functions 7.5 Solving Trigonometric Equations Copyright © 2009 Pearson Education, Inc. 7.2 Identities: Cofunction, Double-Angle, Half-Angle Use cofunction identities to derive other identities. Use the double-angle identities to find function values of twice an angle when one function value is known for that angle. Use the half-angle identities to find function values of half an angle when one function value is known for that angle. Simplify trigonometric expressions using the doubleangle identities and the half-angle identities. Copyright © 2009 Pearson Education, Inc. Cofunction Identities sin x cos x 2 cos x sin x 2 tan x cot x 2 cot x tan x 2 sec x csc x 2 csc x sec x 2 Copyright © 2009 Pearson Education, Inc. Slide 7.2 - 4 Example Prove the identity sin x cos x. 2 Solution: sin x sin x cos cos x sin 2 2 2 sin x 0 cos x 1 cos x Copyright © 2009 Pearson Education, Inc. Slide 7.2 - 5 Cofunction Identities sin x cos x 2 sin x cos x 2 cos x sin x 2 cos x sin x 2 Copyright © 2009 Pearson Education, Inc. Slide 7.2 - 6 Example Find an identity for each of the following. a) tan x b) sec x 90º 2 Solution: sin x cos x 2 a) tan x cot x 2 sin x cos x 2 b) sec x 90º Copyright © 2009 Pearson Education, Inc. 1 1 csc x cos x 90º sin x Slide 7.2 - 7 Double-Angle Identities sin 2x 2sin x cos x cos 2x cos 2 x sin 2 x 1 2sin 2 x 2 cos 2 x 1 2 tan x tan 2x 1 tan 2 x Copyright © 2009 Pearson Education, Inc. Slide 7.2 - 8 Other Useful Identities 1 cos 2x sin x 2 2 1 cos 2x cos x 2 2 1 cos 2x tan x 1 cos 2x 2 Copyright © 2009 Pearson Education, Inc. Slide 7.2 - 9 Example Find an equivalent expression for each of the following. a) sec 3 in terms of function values of 3 b) cos x in terms of function values of x or 2x raised only to the first power Solution: a) sin 3 sin 2 sin 2 cos cos 2 sin 2sin cos cos 2 cos 2 1 sin 2sin cos 2sin cos sin 2 2 4 sin cos2 sin Copyright © 2009 Pearson Education, Inc. Slide 7.2 - 10 Example Solution continued: b) cos3 x cos2 x cos x 1 cos 2x cos x 2 cos x cos x cos 2x 2 Copyright © 2009 Pearson Education, Inc. Slide 7.2 - 11 Half-Angle Identities x 1 cos x sin 2 2 x 1 cos x cos 2 2 Copyright © 2009 Pearson Education, Inc. x 1 cos 2x tan 2 1 cos 2x sin x 1 cos x 1 cos x sin x Slide 7.2 - 12 Example Find tan (π/8) exactly. Then check the answer using a graphing calculator in RADIAN mode. Solution: tan sin 2 2 2 2 2 2 2 4 tan 4 8 2 2 1 cos 1 4 2 2 2 2 2 2 1 2 2 2 2 2 2 Copyright © 2009 Pearson Education, Inc. Slide 7.2 - 13 Example Simplify each of the following. sin x cos x 2 x b) 2sin cos x a) 1 2 cos 2x 2 Solution: sin 2x sin x cos x 2 sin x cos x 2sin x cos x a) 1 2 1 cos 2x cos 2x cos 2x cos 2x 2 2 tan2x 1 cos x 2 x b) 2sin cos x 2 cos x 2 2 1 cos x cos x 1 Copyright © 2009 Pearson Education, Inc. Slide 7.2 - 14