Trigonometric Identities

1. 5 Trigonometric Identities

2. 5 Trigonometric Identities • 5.1 Fundamental Identities • 5.2 Verifying Trigonometric Identities • 5.3 Sum and Difference Identities for Cosine • 5.4 Sum and Difference Identities for Sine and Tangent • 5.5 Double-Angle Identities • 5.6 Half-Angle Identities

3. Sum and Difference Identities for Sine and Tangent 5.4 Sum and Difference Identities for Sine ▪ Sum and Difference Identities for Tangent ▪ Applying the Sum and Difference Identities ▪ Verifying an Identity

4. Sum and Difference Identities for Sine • We can use the cosine sum and difference identities to derive similar identities for sine and tangent. • Cofunction identity • Cosine difference identity • Cofunction identities

5. Sum and Difference Identities for Sine • Sine sum identity • Negative-angle identities

6. Sine of a Sum or Difference

7. Sum and Difference Identities for Tangent • Fundamental identity • Sum identities • Multiply numerator and denominator by 1.

8. Sum and Difference Identities for Tangent • Replace B with –B and use the fact that tan(–B) = –tan B to obtain the identity for the tangent of the difference of two angles. • Multiply. • Simplify. • Fundamental identity

9. Tangent of a Sum or Difference

10. FINDING EXACT SINE AND TANGENT FUNCTION VALUES Example 1(a) Find the exact value of sin 75.

11. FINDING EXACT SINE AND TANGENT FUNCTION VALUES Example 1(b) Find the exact value of

12. FINDING EXACT SINE AND TANGENT FUNCTION VALUES Example 1(c) Find the exact value of

13. WRITING FUNCTIONS AS EXPRESSIONS INVOLVING FUNCTIONS OF θ Example 2 Write each function as an expression involving functions of θ. (a) (b) (c)

14. FINDING FUNCTION VALUES AND THE QUADRANT OF A + B Example 3 Suppose that A and B are angles in standard position withFind each of the following.

15. FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued) Example 3 • The identity for sin(A + B) involves sin A, cos A, sin B, and cos B. The identity for tan(A + B) requires tan A and tan B. We must find cos A, tan A, sin B and tan B. Because A is in quadrant II, cos A is negative and tan A is negative.

16. FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued) Example 3 Because B is in quadrant III, sin B is negative and tan B is positive.

17. FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued) Example 3 (b) (a)

18. FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued) Example 3 • (c) From parts (a) and (b), sin (A + B) > 0 and tan (A + B) > 0. The only quadrant in which the values of both the sine and the tangent are positive is quadrant I, so (A + B) is in quadrant I.

19. VERIFYING AN IDENTITY USING SUM AND DIFFERENCE IDENTITIES Example 4 • Verify that the equation is an identity.