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5.1

Goal : Use the fundamental identities to simplify trigonometric expressions. 5.1. Fundamental Identities. What you’ll learn about. Identities Basic Trigonometric Identities Pythagorean Identities Cofunction Identities Odd-Even Identities Simplifying Trigonometric Expressions

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5.1

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  1. Goal: Use the fundamental identities to simplify trigonometric expressions. 5.1 Fundamental Identities

  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.

  3. FYI

  4. Basic Trigonometric Identities

  5. Pythagorean Identities

  6. Cofunction Identities

  7. Even-Odd Identities

  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

  9. Simplify by Rewriting

  10. Simplify by Expanding Products

  11. Simplifying by Factoring

  12. Simplify by Taking Square Roots

  13. Simplify by Combining Fractions

  14. Example Simplifying by Expanding and Using Identities

  15. Goal: Solve trigonometric equations. 5.1 Day 2 Fundamental Identities

  16. Warm Up Simplify.

  17. Example 1: Using the Pythagorean Identity Given , use the Pythagorean Identity to find the exactly.

  18. Example 2a: Solving Trigonometric Equations by Factoring Find all solutions to the trigonometric equation below.

  19. Example 2b: Solving Trigonometric Equations by Factoring Find all solutions to the trigonometric equation below.

  20. Example 2c: Solving Trigonometric Equations by Factoring Find all solutions to the trigonometric equation in the interval [0, 2π).

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