Section 7.6
Double-angle and Half-angle
Formulas
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Copyright © 2013 Pearson Education, Inc. All rights reserved
Copyright © 2013 Pearson Education, Inc. All rights reserved
2
3π
− ,π < θ <
If cosθ =
, find the exact value of:
5
2
(a) sin ( 2θ )
(b) cos ( 2θ )
y = 52
− 22 =
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) cos2
=
θ cos 2 θ − sin 2 θ
−1
21
−2
5
−3
2
21 
4 21
17
 2 
= −  −  −
=−
 = −
25 25
25
 5  5 
2
−4
−5
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Copyright © 2013 Pearson Education, Inc. All rights reserved
Copyright © 2013 Pearson Education, Inc. All rights reserved
Copyright © 2013 Pearson Education, Inc. All rights reserved
Copyright © 2013 Pearson Education, Inc. All rights reserved
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=
θ
) cos 3=
2
1
cos 2θ cos θ − sin 2θ sin=
θ cos ( 2θ + θ=
θ
) cos 3=
2
2π
π
3θ =
+ 2kπ or 3θ =
+ 2 kπ
3
3
π 2 kπ
2π 2kπ
θ=
+
or θ =
+
9
3
9
3
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Copyright © 2013 Pearson Education, Inc. All rights reserved
Copyright © 2013 Pearson Education, Inc. All rights reserved
Copyright © 2013 Pearson Education, Inc. All rights reserved
Use a Half-angle Formula to find the exact value of:
(b) cos5π
(a) sin 22.5°
12
45°
(a) sin=
22.5° sin
=
2
5π
5π
6
(b) cos
= cos
=
12
2
3
1−
2
=
=
2
2
1−
1 − cos 45°
2
=
2
2
5π
1 + cos
6
2
2− 3
=
4
2− 2
=
4
2− 2
=
2
2− 3
2
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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
=
2
5
5
5
2
 1
α
2
2
10
1+  − 
(b) cos = 1 + cos α
= = =
5


2
=
2
5
5
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
Copyright © 2013 Pearson Education, Inc. All rights reserved
Copyright © 2013 Pearson Education, Inc. All rights reserved