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Math.1330 – Section 6.
Double and Half Angle Formulas
We know trigonometric values of many angles on the unit circle.
Can we use them to find values for more angles?
For example, we know all trigonometric values of 45°;
can we use that information to find trigonometric values of 22.5°?
Or, if we know that sin(𝑥𝑥
1
3
, is there a way to find sin(2𝑥𝑥)?
Yes, there is a way evaluating half/double of the angles we know.
Before introducing such formulas that allows us to evaluate different angles,
let’s emphasize the followings:
sin(2𝑥𝑥) ≠ 2 sin(𝑥𝑥)
cos(2𝑥𝑥) ≠ 2cos(𝑥𝑥)
tan(2𝑥𝑥
≠ 2tan(𝑥𝑥)
To see this, we have:
sin(30°) =
1
2
→ 2 ∙ sin(30°) = 2 ∙
1
2
sin(2 ∙ 30°) = sin(60°) =
√
2
sin(2 ∙ 30°) ≠ 2 ∙ sin(30°)
Double Angle Formulas for Sine, Cosine and Tangent.
Sine Formula:
sin(2𝐴𝐴
) = sin( 𝐴𝐴 + 𝐴𝐴)
= sin(𝐴𝐴
) cos( 𝐴𝐴
) + sin( 𝐴𝐴
) cos( 𝐴𝐴)
= 2 sin(𝐴𝐴
) cos( 𝐴𝐴)
Cosine Formula:
cos(2𝐴𝐴) = cos(𝐴𝐴 + 𝐴𝐴)
= cos(A) cos(𝐴𝐴) − sin(𝐴𝐴) sin(𝐴𝐴)
2
2
cos(2𝐴𝐴) = 𝑐𝑐𝑐𝑐𝑐𝑐
2
2
2
2
Tangent Formula:
tan(2𝐴𝐴) = tan(𝐴𝐴 + 𝐴𝐴)
𝑡𝑡𝑡𝑡𝑠𝑠(𝐴𝐴)+tan(𝐴𝐴)
1 −tan (𝐴𝐴) tan(𝐴𝐴)
2 𝑡𝑡an(𝐴𝐴)
1 −tan
2
(𝐴𝐴)
Double-Angle Formulas:
sin(2𝑥𝑥) = 2 sin(𝑥𝑥) cos(𝑥𝑥)
cos(2𝑥𝑥) = 𝑐𝑐𝑐𝑐𝑐𝑐
2
2
2
2
tan(2𝑥𝑥
2tan(𝑥𝑥)
2
Example 5. Given tan(𝑥𝑥
and 0 < 𝑥𝑥 <
𝜋𝜋
2
, find sin(2𝑥𝑥
Example 6. Given cos(𝜃𝜃
2
3
and 𝜋𝜋 < 𝜃𝜃 <
3𝜋𝜋
2
, evaluate:
a) cos(2𝜃𝜃
b) sin(2𝜃𝜃
c) tan (2𝜃𝜃)
Example 7. In the triangle 𝐴𝐴𝐵𝐵𝐵𝐵 with the right angle 𝐵𝐵, sin(𝐵𝐵) =
1
4
and
|AD|=|BD|. Find sin(𝐴𝐴𝐴𝐴𝐵𝐵
Example 8: Simplify each:
a. 2 sin(75°) cos(75°)
b.
c.
d.
e.
10 sin (𝑥𝑥)cos(𝑥𝑥)
𝑐𝑐𝑐𝑐𝑐𝑐
2
(𝑥𝑥)−𝑐𝑐𝑠𝑠𝑠𝑠
2
(𝑥𝑥)
f.
1 −2 sin
2
(𝑥𝑥)
3 𝑐𝑐𝑐𝑐𝑐𝑐
2
(𝑥𝑥)+3𝑐𝑐𝑠𝑠𝑠𝑠
2
(𝑥𝑥)
g. (2 sin(𝑥𝑥) − 2 cos(𝑥𝑥))
2
sin
cos
2 2
π π
2
2 tan
1 tan 15
1 2 sin ( 6 )
2
− A
Note: In the half-angle formulas the ± symbol is intended to mean either
positive or negative but not both, and the sign before the radical is
determined by the quadrant in which the half-angle
𝑥𝑥
2
terminates.
Example 9: Use a half-angle formula to find the exact value of each.
a.
sin 15 °
b.
cos
c.
tan
π
Example 10: Given sin(𝑥𝑥) = −
4
5
and 𝜋𝜋 < 𝑥𝑥 <
3𝜋𝜋
2
, evaluate:
a) cos �
𝑥𝑥
2
b) sin �
𝑥𝑥
2
c) tan �
𝑥𝑥
2
Example 11. Given tan(𝑥𝑥) = 2 and 0 < 𝑥𝑥 < 𝜋𝜋, find tan(2𝑥𝑥) + tan �
𝑥𝑥
2
Example 12. Given 0 < 𝑥𝑥 < 𝜋𝜋/20, simplify the following
12cos(5𝑥𝑥)
1 − cos(10𝑥𝑥)
