Ch 5.5: Multiple-Angle and Product-to-Sum Formulas

Double Angle Formulassin 2 2sin cosx x x

2

2 tantan 2

1 tan

xx

x

2 2

2

2

cos 2 cos sin

2cos 1

1 2sin

=

=

x x x

x

x

1 cossin2 2

x x

Half Angle Formulas

1 coscos2 2

x x

1 cos sintan2 sin 1 cos

x x x

x x

The + or – depends inwhich quadrant the originalgiven value exists

Ex: Find the sin2Ө and cos2Ө if 5 3

cos , 213 2

x

1. Draw a triangle2. Find the missingpiece3. Use the formula

12sin

13

x

sin 2 2sin cosx x x

2 12 5

1 13 13

120

169

2 2cos 2 cos sinx x x 2 2

5 12

13 13

25 144

169

119

169

5cos

13x

Ex: Find the triple angle formula for: sin 3x

sin 2x x1. Rewrite the inside as a sum

2. Use the formula from 5.4

sin 2 cos cos 2 sinx x x x 3. Replace with double angleformulas

2(2sin cos )cos 1 2sin sinx x x x x 2 32sin cos sin 2sinx x x x 4. Pythagorean Identity

2 32sin 1 sin sin 2sinx x x x 3 32sin 2sin sin 2sinx x x x 5. Simplify

3sin 3 3sin 4sinx x x

Ex: Use the half angle formula to find the exact value of sin(105o)

210sin105 sin

2

2. Double 105 to get the numerator

3. Plug into the half formula

210 1 cos 210sin

2 2

1. 105 exists in the II, so sine is positive!

4.7 3

cos 210 cos6 2

31

22

1 3

2 4

5. Simplify

2 3

4

2 3

2

Product-to-Sum and Sum-to-Product formulas:

Ex: Use the correct formula to write the following product as a sum or difference: cos5 sin 4x x

1cos sin sin( ) sin( )

2x y x y x y

1. Change using formula

1sin(5 4 ) sin(5 4 )

2x x x x

2. Simplify

1sin(9 ) sin( )

2x x

1 1sin(9 ) sin( )2 2

x x

Ex: Find the exact value of cos195 cos105

1. Change using sum-to-product formula

x ycos cos 2cos cos

2 2

x yx y

195 105 195 1052cos cos

2 2

2. Simplify

2cos 150 cos 45

3. Use trig to change cosines

3 22

2 2

6

2

4. Simplify