Holt Algebra 2

14-4 Sum and Difference Identities

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

Holt Algebra 2

14-4 Sum and Difference Identities

Holt Algebra 2

14-4 Sum and Difference IdentitiesExample 1A: Evaluating Expressions with Sum and

Difference IdentitiesFind the exact value of cos 15°.

cos 15° = cos (45° – 30°)

= cos 45° cos 30° + sin 45° sin 30°

Write 15° as the difference 45° – 30° because trigonometric values of 45° and 30° are known.

Apply the identity for cos (A – B).

Evaluate.

Simplify.

Holt Algebra 2

14-4 Sum and Difference IdentitiesExample 1B: Proving Evaluating Expressions with Sum

and Difference Identities

Find the exact value of .

Apply the identity for tan (A + B).

Write as the sum of

Holt Algebra 2

14-4 Sum and Difference IdentitiesExample 1B Continued

Evaluate.

Simplify.

Holt Algebra 2

14-4 Sum and Difference IdentitiesCheck It Out! Example 2

Prove the identity .

= –sin x

Apply the identity for cos A + B.

Evaluate.

Simplify.

Holt Algebra 2

14-4 Sum and Difference IdentitiesCheck It Out! Example 1b

Find the exact value of each expression.

Apply the identity for sin (A – B).

Write as the sum of because

trigonometric values of and are known.

Holt Algebra 2

14-4 Sum and Difference IdentitiesCheck It Out! Example 1b Continued

Find the exact value of each expression.

Evaluate.

Simplify.

Holt Algebra 2

14-4 Sum and Difference IdentitiesExample 3: Using the Pythagorean Theorem with Sum and

Difference Identities

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.

Find cos (A – B) if sin A = with 0 < A < and if tan B = with 0 < B <

Holt Algebra 2

14-4 Sum and Difference IdentitiesExample 3 Continued

x

r = 3y = 1

A

x = 4

y = 3r

B

In Quadrant l (Ql), 0° < A < 90° and sin A = .

In Quadrant l (Ql), 0°< B < 90° and tan B = .

Holt Algebra 2

14-4 Sum and Difference Identities

x2 + 12 = 32

32 + 42 = r2

Example 3 Continued

x

r = 3y = 1

A

x = 4

y = 3r

B

Thus, cos A = Thus, cos B = and sin B = .and sin A =

Holt Algebra 2

14-4 Sum and Difference IdentitiesExample 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).

cos(A – B) =

Substitute for cos A, for cos

B, and for sin B. Simplify.

Holt Algebra 2

14-4 Sum and Difference IdentitiesCheck 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

r = 5y = 4A

x = 3

yr = 5

B

Holt Algebra 2

14-4 Sum and Difference Identities

x

r = 5y = 4A

x = 3

yr = 5

B

Check It Out! Example 3 Continued

52 – 32 = y2

Thus, cos B = and sin B =

x2 + 42 = 52

Thus, sin A =

and cos A =

Holt Algebra 2

14-4 Sum and Difference IdentitiesCheck 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).

Simplify.sin(A – B) =

Substitute for sin A and sin B, for cos A, and for cos B.

Holt Algebra 2

14-4 Sum and Difference IdentitiesLesson 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<