Sum and Difference Identities 14-4 Sum and Difference Identities Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Warm Up Find each product, if possible. 1. AB 2. BA

Objectives Evaluate trigonometric expressions by using sum and difference identities. Use matrix multiplication with sum and difference identities to perform rotations.

Vocabulary rotation matrix

Matrix multiplication and sum and difference identities are tools to find the coordinates of points rotated about the origin on a plane.

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 1a Find the exact value of tan 105°. Write 105° as the sum of 60° + 45° because trigonometric values of 60° and 45° are known. tan 105°= tan(60° + 45°) Apply the identity for tan (A + B).

Check It Out! Example 1a Continued Evaluate. = 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.

Shifting the cosine function right  radians is equivalent to reflecting it across the x-axis. A proof of this is shown in Example 2 by using a difference identity.

Example 2: Proving Identities with Sum and Difference Identities Prove the identity tan tan Choose the left-hand side to modify. Apply the identity for tan (A + B). Evaluate. Simplify.

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

To rotate a point P(x,y) through an angle θ use a rotation matrix. The sum identities for sine and cosine are used to derive the system of equations that yields the rotation matrix.

Example 4: Using a Rotation Matrix Find the coordinates, to the nearest hundredth, of the points (1, 1) and (2, 0) after a 40° rotation about the origin. Step 1 Write matrices for a 40° rotation and for the points in the question. Rotation matrix. Matrix of point coordinates.

Example 4 Continued Step 2 Find the matrix product. Step 3 The approximate coordinates of the points after a 40° rotation are (0.12, 1.41) and (1.53, 1.29).

Check It Out! Example 4 Find the coordinates, to the nearest hundredth, of the points in the original figure after a 60° rotation about the origin. Step 1 Write matrices for a 60° rotation and for the points in the question. R60° = Rotation matrix. S = Matrix of point coordinates.

Check It Out! Example 4 Continued Step 2 Find the matrix product. R60° x s =

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<

Lesson Quiz: Part II 4. Find the coordinates to the nearest hundredth of the point (3, 4) after a 60° rotation about the origin.