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Copyright © 2011 Pearson, Inc. Goal: Use the fundamental identities to simplify trigonometric expressions. 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 - 3 FYI Copyright © 2011 Pearson, Inc. Basic Trigonometric Identities Reciprocal Identites 1 csc sin 1 sec cos 1 cot tan 1 sin csc 1 cos sec 1 tan cot Quotient Identites sin tan cos Copyright © 2011 Pearson, Inc. cos cot tan Slide 5.1 - 5 Pythagorean Identities Copyright © 2011 Pearson, Inc. Slide 5.1 - 6 Cofunction Identities sin cos 2 cos sin 2 tan cot 2 cot tan 2 sec csc 2 csc sec 2 Copyright © 2011 Pearson, Inc. Slide 5.1 - 7 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 Copyright © 2011 Pearson, Inc. Slide 5.1 - 8 Simplification Ideas Rewrite tan, cot, sec, and csc in terms of sin and cos. Expand Products Example: (1 + sin x)(1 – sin x) = 1 – sin x + sin x – sin2x Factor Example: sinx•cosx – sinx = sinx(cosx – 1) Take Square Roots Combining Fractions Copyright © 2011 Pearson, Inc. Simplify by Rewriting tan 𝑥 ∙ cos 𝑥 Copyright © 2011 Pearson, Inc. Slide 5.1 - 10 Simplify by Expanding Products sin 𝑥 tan 𝑥 + cot 𝑥 Copyright © 2011 Pearson, Inc. Slide 5.1 - 11 Simplifying by Factoring Simplify the expression cos3 x cos xsin2 x. Copyright © 2011 Pearson, Inc. Slide 5.1 - 12 Simplify by Taking Square Roots 1 + tan2 𝜃 Copyright © 2011 Pearson, Inc. Slide 5.1 - 13 Simplify by Combining Fractions sec 𝑥 sin 𝑥 − sin 𝑥 cos 𝑥 Copyright © 2011 Pearson, Inc. Slide 5.1 - 14 Example Simplifying by Expanding and Using Identities csc x -1csc x 1 Simplify the expression: cos 2 x Copyright © 2011 Pearson, Inc. Slide 5.1 - 15 Goal: Solve trigonometric equations. 5.1 Day 2 Fundamental Identities Copyright © 2011 Pearson, Inc. Warm Up Simplify. 1 − sin2 𝜃 cos 𝜃 Copyright © 2011 Pearson, Inc. Slide 5.1 - 17 Example 1: Using the Pythagorean Identity Given sin 𝜃 = exactly. Copyright © 2011 Pearson, Inc. 3 , 4 use the Pythagorean Identity to find the cos 𝜃 Slide 5.1 - 18 Example 2a: Solving Trigonometric Equations by Factoring Find all solutions to the trigonometric equation below. 2 sin2 𝑥 + sin 𝑥 − 1 = 0 Copyright © 2011 Pearson, Inc. Slide 5.1 - 19 Example 2b: Solving Trigonometric Equations by Factoring Find all solutions to the trigonometric equation below. 2 sin2 𝑥 + 3sin 𝑥 = 2 Copyright © 2011 Pearson, Inc. Slide 5.1 - 20 Example 2c: Solving Trigonometric Equations by Factoring Find all solutions to the trigonometric equation in the interval [0, 2π). tan 𝑥 sin2 𝑥 = tan 𝑥 Copyright © 2011 Pearson, Inc. Slide 5.1 - 21