# 14-4 Sum and Difference Identities Check it out 14-4 Sum and Difference Identities Check it out Does the sin(75) =sin(45)+sin(30) ? Holt Algebra 2 14-4 Sum and Difference Identities Holt Algebra 2 14-4 Sum and Difference Identities Example 1A: Evaluating Expressions with Sum and Difference Identities Find the exact value of cos 15. Write 15 as the difference 45 30 because

cos 15 = cos (45 30) trigonometric values of 45 and 30 are known. Apply the identity for cos (A B). = cos 45 cos 30 + sin 45 sin 30 Evaluate. Simplify. Holt Algebra 2 14-4 Sum and Difference Identities 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). Holt Algebra 2 14-4 Sum and Difference Identities Example 1B Continued Evaluate. Simplify. Holt Algebra 2 14-4 Sum and Difference Identities

Check It Out! Example 2 Prove the identity . Apply the identity for cos A + B. Evaluate. = sin x Holt Algebra 2 Simplify. 14-4 Sum and Difference Identities 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). Holt Algebra 2 14-4 Sum and Difference Identities Check It Out! Example 1b Continued Find the exact value of each expression. Evaluate.

Simplify. Holt Algebra 2 14-4 Sum and Difference Identities Example 3: Using the Pythagorean Theorem with Sum and Difference Identities Find cos (A B) if sin A = if tan B = with 0 < A < and 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. Holt Algebra 2 14-4 Sum and Difference Identities Example 3 Continued In Quadrant l (Ql), 0 < A < 90 and sin A = . r=3 A x

Holt Algebra 2 In Quadrant l (Ql), 0< B < 90 and tan B = . r y=1 B x=4 y=3 14-4 Sum and Difference Identities Example 3 Continued r=3 A

r y=3 y=1 x B x=4 x2 + 12 = 32 32 + 42 = r2 Thus, cos A = and sin A = Holt Algebra 2

Thus, cos B = sin B = . and 14-4 Sum and Difference Identities 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 cos(A B) = Holt Algebra 2 Simplify. for sin B. 14-4 Sum and Difference Identities 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 = r=5 y=4 A x Holt Algebra 2 y r=5 B

x=3 14-4 Sum and Difference Identities Check It Out! Example 3 Continued r=5 y=4 y r=5 A x B x=3 x2 + 42 = 52

52 32 = y2 Thus, sin A = Thus, cos B = and cos A = sin B = Holt Algebra 2 and 14-4 Sum and Difference Identities 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) = Holt Algebra 2 Simplify. 14-4 Sum and Difference Identities

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