Double-Angle and Half-angle Formulas

2 2

2

2

2

sin(2 ) 2sin cos

cos(2 ) cos sin

cos(2 ) 1 2sin

cos(2 ) 2cos 12 tantan(2 )

1 tan

3If sin , , find the exact value of:5 2

( ) sin(2 ) ( ) cos 2 ( ) tan 2a b c

2 2 2

2 2 2

2

2

sin 2 2sin cos

3sin 3 5. We need to find 5

5 3

25 9

164 ( 2 )

4cos5

y so y and r xr

r x y

x

x

xx Negative because the angle is in the nd quadrant

2

2

3 4( ) sin 2 25 5

2425

( ) cos 2 1 2sin ( sin )

31 259 71 2 ( ? ?)25 25

a

b Use this formula because is given

Why is it positive What quad

2

2

2 tan 3( ) tan 2 tan1 tan 4

324314

3291

163

3 16 2427 2 7 7

16

ycx

Develop a formula for sin (3θ) in terms of sin θ and cos θ.

2 2

2 2 3

2 3

sin(3 ) sin(2 ) sin 2 cos cos 2 sin

2sin cos cos cos sin sin

2sin cos sin cos sin

3sin cos sin

2

2

2

1 cos 2sin

2

1 cos 2cos

2

1 cos 2tan

1 cos 2

Write an equivalent expression for cos^4θ that does not involve any powers of sine or cosine greater than 1.

24 2 2

2

2

1 cos(2 )cos (cos )2

1 2cos 2 cos (2 )4

1 1 1cos(2 ) cos (2 )4 2 41 1 1 1 cos(2(2 )cos(2 )4 2 4 21 1 1cos(2 ) (1 cos(4 ))4 2 81 1 1 1 3 1 1cos(2 ) (cos 4 ) cos 2 cos(4 )4 2 8 8 8 2 8

1 cossin2 2

1 coscos2 2

1 costan2 1 cos

Find the exact value of cos (15 degrees)

Find the exact value of sin (165 deg)

Find the exact value of tan (195 deg)

Remember to be careful to check the sign of the cosine of the angle

Remember to look at the quadrant to determine if the final answer is positive or negative

3 3If cos , , find the exact value of 5 2

( ) sin ( ) cos ( ) tan2 2 2

a b c

3 818 1 25 5( ) sin

2 2 2 5 2 5( ?)

3 212 1 15 5( ) cos

2 2 2 5 2 52

2 55( ) tan 212 155

a

Why positive

b

c

1 cos sintan2 sin 1 cos

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