Memorization Quiz

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

Warm Up

Prove the cofunction identity:

Section/Topic 5.4a Sum/Difference Identities for Sine and Tangent.

CC High School Functions

Trigonometric Functions: Prove and Apply trigonometric identities

Objective Students will be able to verify identities using all the previously learned ID and the Sum/Diff ID for Sine and Tangent.

Homework Page 214-216 (3-8 all, 9-65 odd) due Tue, 12/10Quiz 5.3 to 5.4 next Tuesday 12/10/13

Trig Game Plan Date: 12/06/13

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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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Sum and Difference Identities for Sine

Sine sum identity

Negative-angle identities

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1.1-6

5.4-6

Sine of a Sum or Difference

sin ( A ± B ) = sin A cos B ± cos A sin B

Sign stays Same

Example 1(a) FINDING EXACT SINE AND TANGENT FUNCTION VALUES

Find the exact value of sin 75.

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Find the exact value of each expression.

(a)

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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

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)