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5.1 Using Fundamental Identities Quick Review Evaluate the expression. 4 1. sin 5 12 2. cos 13 Factor the expression into a product of linear factors. -1 -1 3. 2a 3ab 2b 2 2 4. 9u 6u 1 2 Simplify the expression. 5. 2 3 y x Quick Review Solutions Evaluate the expression. 4 1. sin 53.13 0.927 rad 5 12 2. cos 157.38 2.747 rad 13 Factor the expression into a product of linear factors. -1 -1 3. 2a 3ab 2b 2 2 4. 9u 6u 1 2 2a b a 2b 3u 1 Simplify the expression. 5. 2 3 y x 2x 3y xy 2 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. Basic Trigonometric Identities Reciprocal Identites 1 csc sin sin 1 csc 1 sec cos cos 1 sec Quotient Identites sin tan cos cos cot tan 1 cot tan tan 1 cot Pythagorean Identities cos sin 1 2 2 1 tan sec 2 2 cot 1 csc 2 2 Example Using Identities Find sin and cos if tan 3 and cos 0. Example Using Identities Find sin and cos if tan 3 and cos 0. 1 tan sec 2 2 1 9 sec since tan 3 2 sec 10 cos 1/ 10 since cos 0 To find sin , use tan 3 and cos 1/ 10. sin cos sin cos tan tan sin 1/ 10 3 sin 3/ 10 Therefore, cos 1/ 10 and sin 3/ 10 Cofunction Identities y r x cos A r x Angle B: sin B r y cos B r Angle A: sin A y x x cot A y x tan B y y cot B x tan A r x r csc A y r sec B y r csc B x sec A Cofunction Identities sin cos 2 cos sin 2 tan cot 2 cot tan 2 sec csc 2 csc sec 2 Even-Odd Identities sin(- x) -sin x cos(- x) cos x tan(- x) - tan x csc(- x) - csc x sec(- x) sec x cot(- x) - cot x Example Simplifying by Factoring and Using Identities Simplify the expression cos x cos x sin x. 3 2 Example Simplifying by Factoring and Using Identities Simplify the expression cos x cos x sin x. 3 cos x cos x sin x 3 2 cos x(cos x sin x) 2 cos x(1) cos x 2 Pythagorean Identity 2 Example Simplifying by Expanding and Using Identities Simplify the expression: csc x -1 csc x 1 2 cos x Example Simplifying by Expanding and Using Identities Simplify the expression: csc x -1 csc x 1 csc x -1 csc x 1 2 cos x 2 cos x csc x 1 cos x cot x cos x cos x 1 sin x cos x 1 sin x csc x 2 2 ( a b)(a - b) a - b 2 2 2 2 2 2 2 2 2 Pythagorean Identity cos cot sin Example Solving a Trigonometric Equation 3 sin x Find all values of x in the interval [0,2 ) that solve tan x. cos x Example Solving a Trigonometric Equation 3 sin x Find all values of x in the interval [0,2 ) that solve tan x. cos x 3 sin x tan x cos x sin x sin x cos x cos x sin x sin x 3 3 sin x sin x 0 3 sin x(sin x 1) 0 2 sin x cos x 0 2 sin x 0 or cos x 0 2 Reject the posibility that cos x 0 because it would make both 2 sides of the original equation undefined. sin x 0 in the interval 0 x 2 when x 0 and x . Homework