Section 10.5: Multiple-Angle and Product-Sum Formulas
The following identities are provided without proof. You DO NOT need to memorize
them for the test, they will be provided. You only need to know how to use them.
Double Angle Formulas:
Power Reducing Formulas:
sin( 2u ) = 2 sin u cos u
sin 2 u =
1 − cos 2u
2
cos(2u ) = cos 2 u − sin 2 u
= 2 cos 2 u − 1
= 1 − 2 sin 2 u
cos 2 u =
1 + cos 2u
2
tan(2u ) =
2 tan u
1 − tan 2 u
tan 2 (u ) =
1 − cos 2u
1 + cos 2u
Half Angle Formulas:
sin
u
1 − cos u
=±
2
2
cos
u
1 + cos u
=±
2
2
Note: These formulas are VERY SPECIFIC.
trigonometric function:
sin(2u ) ≠ 2 sin u
Ex 1:
17
1
θ
4
Find: sin(2θ )
8
⎛ 1 ⎞⎛ 4 ⎞
sin( 2θ ) = 2 sin θ cos θ = 2⎜⎜
⎟⎟⎜⎜
⎟⎟ =
17
⎝ 17 ⎠⎝ 17 ⎠
You CAN NOT distribute ANY
Ex 2: Solve on the interval [0, 2π)
cos 2 x + sin x = 0
To solve this we need to convert to all sine or all cosine. We will convert the cos 2x term
to sines for this problem using the following identity:
cos(2u ) = 1 − 2 sin 2 u
So
cos 2 x + sin x = 0
1 − 2 sin 2 x + sin x = 0
(1 − sin x )(1 + 2 sin x ) = 0
We have two equations now:
1 − sin x = 0
sin x = 1
OR
x = π/2
1 + 2 sin x = 0
sin x = −1 / 2
x = 7π/6, 11π/6
Ex 3: Rewrite the expression using a double angle formula:
6 cos 2 x − 3
6 cos 2 x − 3 = 3(2 cos 2 x − 1)
= 3 cos 2 x
2
and π/2 ≤ u ≤ π
3
We first need to find u so we have to draw a triangle in the second quadrant:
Ex 4: Find sin 2u if cos u = −
3
u
5
-2
Then we have to use the identity:
4 5
⎛ 5 ⎞⎛ 2 ⎞
⎟⎜ − ⎟ = −
sin( 2u ) = 2 sin u cos u = 2⎜⎜
⎟
9
⎝ 3 ⎠⎝ 3 ⎠
Ex 5: Rewrite sin 4 ( x ) using power reducing formulas. We want an expression with no
exponents larger than 1.
sin 4 ( x ) = (sin 2 x )
2
2
⎛ 1 − cos 2 x ⎞
=⎜
⎟
2
⎝
⎠
1 − 2 cos 2 x + cos 2 2 x
=
4
We must reduce one more time for cos 2 2 x
⎛ 1 + cos 4 x ⎞
1 − 2 cos 2 x + ⎜
⎟
2
⎝
⎠
=
4
Ex 6: Use a half angle formula to find sin 165
1 − cos 330
⎛ 330 ⎞
sin 165 = sin⎜
=
⎟=+
2
⎝ 2 ⎠
1−
2
3
2 =
2− 3
4