Section 7.6
Double-angle and Half-angle
Formulas
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2
3π
If cosθ = − , π < θ <
, find the exact value of:
5
2
2
(a) sin ( 2θ )
(b) cos ( 2θ )
y= 5
2
− 2 = 21
⎛
21 ⎞ ⎛ 2 ⎞ 4 21
(a) sin 2θ = 2sin θ cos θ = 2 ⎜⎜ −
⎟⎟ ⎜ − ⎟ =
25
⎝ 5 ⎠ ⎝ 5 ⎠
y
2
x
−2
−1
1
2
3
(b) cos2θ = cos2 θ − sin 2 θ
−1
21
−2
5
−3
−4
−5
2
2
21 ⎞
4 21
17
⎛ 2 ⎞ ⎛
= ⎜ − ⎟ − ⎜⎜ −
−
=−
⎟⎟ =
25
⎝ 5 ⎠ ⎝ 5 ⎠ 25 25
1
Solve the equation: cos 2θ cos θ − sin 2θ sin θ =
2
1
cos 2θ cos θ − sin 2θ sin θ = cos ( 2θ + θ ) = cos3θ =
2
1
cos 2θ cos θ − sin 2θ sin θ = cos ( 2θ + θ ) = cos3θ =
2
π
2π
3θ = + 2kπ or 3θ =
+ 2 kπ
3
3
π 2kπ
2π 2kπ
θ= +
or θ =
+
9
3
9
3
Use a Half-angle Formula to find the exact value of:
(a) sin 22.5°
(b) cos 5π
12
2
1−
45°
1 − cos 45°
2
(a) sin 22.5° = sin
=
=
2
2
2
5π
5π
1 + cos
5π
6
(b) cos
= cos 6 =
12
2
2
3
1−
2
=
2
2− 3
2− 3
=
=
4
2
2− 2
=
4
2− 2
=
2
1 π
If cos α = − , < α < π , find the exact value of:
5 2
⎛ 1 ⎞
α
3
3
15
1 − ⎜ − ⎟
(a) sin = 1 − cos α
=
=
=
5
⎝
⎠
2
=
5
5
2
5
2
⎛ 1 ⎞
α
2
2
10
1 + ⎜ − ⎟
(b) cos = 1 + cos α
=
=
=
5
⎝
⎠
2
=
5
5
2
5
2
15
α
sin
α
15
150 5 6
6
5
2
(c) tan
=
=
=
=
=
=
2
α
10
10
2
10
10
cos
2
5
π
2
< α < π , so
π
4
<
α
2
<
π
2
so
α
2
is in quadrant I