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5.1 Fundamental Identities Copyright © 2011 Pearson, Inc. What you’ll learn about Identities Basic Trigonometric Identities Pythagorean Identities Cofunction Identities Odd-Even Identities Simplifying Trigonometric Expressions Solving Trigonometric Equations … and why Identities are important when working with trigonometric functions in calculus. Copyright © 2011 Pearson, Inc. Slide 5.1 - 2 Basic Trigonometric Identities Reciprocal Identites 1 1 csc sec sin cos 1 cot tan 1 sin csc 1 tan cot 1 cos sec Quotient Identites sin cos tan cot cos tan Copyright © 2011 Pearson, Inc. Slide 5.1 - 3 Pythagorean Identities cos sin 1 1 tan sec cot 1 csc 2 2 2 2 Copyright © 2011 Pearson, Inc. 2 2 Slide 5.1 - 4 Example Using Identities Find sin and cos if tan 3 and cos 0. Copyright © 2011 Pearson, Inc. Slide 5.1 - 5 Example Using Identities Find sin and cos if tan 3 and cos 0. 1 tan 2 sec 2 To find sin , use tan 3 1 9 sec 2 and cos 1 / 10. sin tan cos sin cos tan sec 10 cos 1 / 10 sin 1 / 10 3 sin 3 / 10 Therefore, cos 1 / 10 and sin 3 / 10 Copyright © 2011 Pearson, Inc. Slide 5.1 - 6 Cofunction Identities y Angle A: sin A r x cos A r x Angle B: sin B r y cos B r Copyright © 2011 Pearson, Inc. y tan A x x cot A y x tan B y y cot B x r sec A x r csc A y r sec B y r csc B x Slide 5.1 - 7 Cofunction Identities sin cos 2 tan cot 2 sec csc 2 Copyright © 2011 Pearson, Inc. cos sin 2 cot tan 2 csc sec 2 Slide 5.1 - 8 Even-Odd Identities sin(x) sin x csc(x) csc x Copyright © 2011 Pearson, Inc. cos(x) cos x sec(x) sec x tan(x) tan x cot(x) cot x Slide 5.1 - 9 Example Simplifying by Factoring and Using Identities Simplify the expression cos 3 x cos xsin2 x. Copyright © 2011 Pearson, Inc. Slide 5.1 - 10 Example Simplifying by Factoring and Using Identities Simplify the expression cos 3 x cos xsin2 x. cos 3 x cos x sin 2 x cos x(cos2 x sin 2 x) cos x(1) Pythagorean Identity cos x Copyright © 2011 Pearson, Inc. Slide 5.1 - 11 Example Simplifying by Expanding and Using Identities csc x -1csc x 1 Simplify the expression: cos2 x Copyright © 2011 Pearson, Inc. Slide 5.1 - 12 Example Simplifying by Expanding and Using Identities csc x 1csc x 1 csc 2 x 1 cos 2 x Copyright © 2011 Pearson, Inc. (a b)(a b) a 2 b 2 cos 2 x cot 2 x Pythagorean Identity 2 cos x cos 2 x 1 cos cot 2 2 sin x cos x sin 1 sin 2 x csc 2 x Slide 5.1 - 13 Example Solving a Trigonometric Equation Find all values of x in the interval 0,2 sin 3 x that solve tan x. cos x Copyright © 2011 Pearson, Inc. Slide 5.1 - 14 Example Solving a Trigonometric Equation 3 sin x tan x cos x sin 3 x sin x cos x cos x 3 sin x sin x Reject the posibility that cos 2 x 0 because it would make both sides of the original equation undefined. sin x 0 in the interval 0 x 2 when x 0 and x . sin 3 x sin x 0 sin x(sin x 1) 0 2 sin x 0 or 2 sin x cos x 0 Copyright © 2011 Pearson, Inc. cos 2 x 0 Slide 5.1 - 15