Section 7.3 Double-Angle, Half-Angle and Product-Sum Formulas

Section 7.3Double-Angle, Half-Angle and

Product-Sum FormulasObjectives:•To understand and apply the double-angle formula.•To understand and apply the half-angle formula.•To understand and apply product-sum formula.

Class Work

1. Evaluate csc780

2. Find the exact value of tan165

Double-Angle Formulas

Formula for sine:

Formula for cosine:

Formula for tangent:

sin 2 2sin cosx x x

2 2cos 2 cos sinx x x

2

2 tantan 2

1 tan

xx

x

21 2sin x 22cos 1x

Ex 1. If and x is in quadrant II, find cos 2x, sin 2x, and tan 2x.

2cos

3x

Ex 2. If and x is in quadrant I, then find sin 2x, cos 2x, and tan 2x.

3sin

5x

Class Work

3. If and x is in quadrant II, find sin 2x, cos 2x, and tan 2x.

6cos

11x

Half-Angle Formulas1 cos

sin2 2

1 coscos2 2

1 costan2 sin

sin

1 cos

u u

u u

u u

uu

u

Ex 3. Find the exact value of sin 22.5.

Ex 4. Find tan8

Ex 5. Find if and u is in quad II. tan2

u 2sin

5u

Class Work

4. Find if and u is in quad IV.

sin , cos , tan2 2 2

u u uand

2cos

3u

Simplify the expression by using a double-angle or half-angle formula.

5.

6.

7.

2sin35 cos35

2 2cos sin2 2

1 cos4

2

Product-to-Sum Formulas

sin u cos v = ½[sin(u + v) + sin(u – v)]

cos u sin v = ½[sin(u + v) – sin(u – v)]

cos u cos v = ½[cos(u + v) + cos(u – v)]

sin u sin v = ½[cos(u – v) – cos(u + v)]

Ex 6. Express as a sum of trig functions.

sin 3 sin 5x x

Ex 7. Compute sin 15 cos 15.

Class Work

8. Write the product as a sum.

9. Compute:

cos3 cos7x x

3cos37.5 cos7.5

Sum-to-Product Formulas

sin sin 2sin cos2 2

sin sin 2cos sin2 2

cos cos 2cos cos2 2

cos cos 2sin sin2 2

x y x yx y

x y x yx y

x y x yx y

x y x yx y

EX 8. Write as a product. sin 7 sin 3x x

Class Work

10. Write as a product. cos11 cos5x x

HW #2 p 548 1-7 odd, 15-23 odd, 27-31 odd, 35-57 odd.