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

3. 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.

4. 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).

5. Example 1B Continued
Evaluate.
Simplify.

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

7. 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).

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

9. 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.

10. 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

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

12. 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) =

13. 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

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

15. 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.

16. 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<