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

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Warm Up

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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 214-216 (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

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Copyright © 2009 Pearson Addison-Wesley 5.4-4 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

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Copyright © 2009 Pearson Addison-Wesley 5.4-5 Sum and Difference Identities for Sine Sine sum identity Negative-angle identities

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Copyright © 2009 Pearson Addison-Wesley1.1-6 5.4-6 Sine of a Sum or Difference sin ( A ± B ) = sin A cos B ± cos A sin B Sign stays Same

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

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

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Copyright © 2009 Pearson Addison-Wesley1.1-9 5.4-9

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

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

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

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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.

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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

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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.

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

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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.

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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

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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)

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

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

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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)

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