Memorization Quiz Reciprocal (6) Pythagorean (3) Quotient (2) Negative Angle (6) Cofunction (6) Cosine Sum and Difference (2)

Warm Up

Section/Topic5.4a Sum/Difference Identities for Sine and Tangent. CC High School Functions Trigonometric Functions: Prove and Apply trigonometric identities ObjectiveStudents will be able to verify identities using all the previously learned ID and the Sum/Diff ID for Sine and Tangent. HomeworkPage (3-8 all, 9-65 odd) due Tue, 12/10 Quiz 5.3 to 5.4 next Tuesday 12/10/13 Trig Game Plan Date: 12/06/13

Copyright © 2009 Pearson Addison-Wesley Sum and Difference Identities for Sine Cofunction identity We can use the cosine sum and difference identities to derive similar identities for sine and tangent. Cosine difference identity Cofunction identities

Copyright © 2009 Pearson Addison-Wesley Sum and Difference Identities for Sine Sine sum identity Negative-angle identities

Copyright © 2009 Pearson Addison-Wesley Sine of a Sum or Difference sin ( A ± B ) = sin A cos B ± cos A sin B Sign stays Same

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-8 Find the exact value of each expression. (a)

Copyright © 2009 Pearson Addison-Wesley

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

Write each function as an expression involving functions of θ. Basic Advanced

Write each function as an expression involving functions of θ. Basic Advanced

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

FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued) The identity for sin(A + B) requires 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. Example 3a

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

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

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

Suppose that A and B are angles in standard position with and Find each of the following. (c)the quadrant of A – B. Example 3b FINDING FUNCTION VALUES AND THE QUADRANT OF A + B

The identity for sin(A – B) requires sin A, cos A, sin B, and cos B. The identity for tan(A – B) requires tan A and tan B. We must find sin A, tan A, cos B and tan B. Because A is in quadrant III, sin A is negative and tan A is positive. Because B is in quadrant IV, cos B is positive and tan B is negative. Example 3b FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued)

To find sin A and cos B, use the identity Example 3b FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued)

To find tan A and tan B, use the identity Example 3b FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued)

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 negative is quadrant IV, so (A − B) is in quadrant IV. Example 3b FINDING FUNCTION VALUES AND THE QUADRANT OF A + B (continued)