5.5 Multiple-Angle and Product-Sum Formulas

Find all solutions in 02sincos2 =+ xx0cossin2cos2 =+ xxx

0)sin1(cos2 =+ xx0cos =x 1sin −=x

2

3,

2

ππ=x

2

3π=x

( )π2,0

Sketch the graph of y = 4 cos2 x - 2 over [ ]π2,0

= 2(2 cos2 x - 1)

= 2 cos 2x

π2π

2

2−

Use the fact that ππ2

2

3,

13

5cos <<= xx

to find sin 2x, cos 2x, and tan 2x

sin 2x = 2 sin x cos x =169

120

13

5

13

122 −=⎟

⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−

cos 2x = 2 cos2 x - 1 =169

1191

169

252 −=−⎟

⎠

⎞⎜⎝

⎛

tan 2x = =x

x

2cos

2sin=

−

−

169119169120

119

120

-12

5

13

Express sin 3x in terms of sin x.

sin 3x = sin (2x + x)

= sin 2x cos x + cos 2x sin x

= 2 sin x cos x cos x + (1 - 2 sin2 x) sin x

= 2 sin x cos2 x + sin x - 2 sin3 x

= 2 sin x(1 - sin2 x) + sin x - 2 sin3 x

= 2 sin x - 2 sin3 x + sin x - 2 sin3 x

= 3 sin x - 4 sin3 x

Rewrite sin4 x as a sum involving first powers of cos x

sin4 x = (sin2x)2 2

2

2cos1⎟⎠

⎞⎜⎝

⎛ −=

x

Foil ( )xx 2cos2cos214

1 2+−=

⎟⎠

⎞⎜⎝

⎛ ++−=

2

4cos12cos21

4

1 xx

xx 4cos8

1

8

12cos

2

1

4

1++−=

xx 4cos8

12cos

2

1

8

3+−=

( )xx 4cos2cos438

1+−=

Find the exact value of sin 105o

sin 105o =2

210cos1 o− sin of 105 is positive

22

31+

= ⎟⎠

⎞⎜⎝

⎛2

2

2

32 +=

=2

210sin

Writing Products as Sums

Rewrite cos 5x sin 4x as a sum or difference.

cos 5x sin 4x = [ ])45sin()45sin(2

1xxxx −−+

xx sin2

19sin

2

1−=

Using a Sum-to-Product Formula

Find the exact value of cos 1950 + cos 1050

cos 1950 + cos 1050

⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

2

105195cos

2

105195cos2

0000

= 2 cos 1500 cos 450

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

2

2

2

32

2

6−=

Solving a Trigonometric Equation

Find all solutions of sin 5x + sin 3x = 0 in

02

35cos

2

35sin2 =⎟

⎠

⎞⎜⎝

⎛ −⎟⎠

⎞⎜⎝

⎛ + xxxx

2sin 4x cos x = 0

sin 4x = 0 cos x = 0

[ )π2,0

4

7,

2

3,

4

5,,

4

3,

2,

4,0

πππππππ=∴x

2

3,

2

ππ=∴xπkx 204 +=

2

πkx=

ππ kx 24 +=

24

ππ kx +=

Plug in 0, 1, 2, and 3