Check it out Does the sin(75) =sin(45)+sin(30) ?

Example 1A: Evaluating Expressions with Sum and Difference Identities Find the exact value of cos 15°. Write 15° as the difference 45° – 30° because trigonometric values of 45° and 30° are known. cos 15° = cos (45° – 30°) Apply the identity for cos (A – B). = cos 45° cos 30° + sin 45° sin 30° Evaluate. Simplify.

Example 1B: Proving Evaluating Expressions with Sum and Difference Identities Find the exact value of . Write as the sum of Apply the identity for tan (A + B).

Example 1B Continued Evaluate. Simplify.

Check It Out! Example 2 Prove the identity . Apply the identity for cos A + B. Evaluate. = –sin x Simplify.

Check It Out! Example 1b Find the exact value of each expression. Write as the sum of because trigonometric values of and are known. Apply the identity for sin (A – B).

Check It Out! Example 1b Continued Find the exact value of each expression. Evaluate. Simplify.

Example 3: Using the Pythagorean Theorem with Sum and Difference Identities Find cos (A – B) if sin A = with 0 < A < and if tan B = with 0 < B < Step 1 Find cos A, cos B, and sin B. Use reference angles and the ratio definitions sin A = and tan B = Draw a triangle in the appropriate quadrant and label x, y, and r for each angle.

In Quadrant l (Ql), 0° < A < 90° and sin A = . Example 3 Continued In Quadrant l (Ql), 0° < A < 90° and sin A = . In Quadrant l (Ql), 0°< B < 90° and tan B = . x = 4 y = 3 r B x r = 3 y = 1 A

Example 3 Continued x2 + 12 = 32 32 + 42 = r2 y = 3 y = 1 A B x x = 4 x2 + 12 = 32 32 + 42 = r2 Thus, cos A = Thus, cos B = and sin B = . and sin A =

Example 3 Continued Step 2 Use the angle-difference identity to find cos (A – B). cos (A – B) = cosAcosB + sinA sinB Apply the identity for cos (A – B). Substitute for cos A, for cos B, and for sin B. Simplify. cos(A – B) =

Check It Out! Example 3 Find sin (A – B) if sinA = with 90° < A < 180° and if cosB = with 0° < B < 90°. In Quadrant ll (Ql), 90< A < 180 and sin A = . In Quadrant l (Ql), 0< B < 90° and cos B = x = 3 y r = 5 B x r = 5 y = 4 A

Check It Out! Example 3 Continued r = 5 y = 4 A x = 3 y r = 5 B x2 + 42 = 52 Thus, sin A = and cos A = 52 – 32 = y2 Thus, cos B = and sin B =

Check It Out! Example 3 Continued Step 2 Use the angle-difference identity to find sin (A – B). sin (A – B) = sinAcosB – cosAsinB Apply the identity for sin (A – B). Substitute for sin A and sin B, for cos A, and for cos B. sin(A – B) = Simplify.

Lesson Quiz: Part I 1. Find the exact value of cos 75° 2. Prove the identity sin = cos θ 3. Find tan (A – B) for sin A = with 0 <A< and cos B = with 0 <B<