4. Half-Angle Formulas
by M. Bourne
We will develop formulas for the sine and cosine of a half angle.
We start with the formula for the cosine of a double angle that we met in the last section.
cos 2θ = 1− 2sin2 θ
Half Angle Formula - Sine
Now, if we let
then 2θ = α and our formula becomes:
cos α = 1 − 2sin2 (α/2)
We now solve for
(that is, we get sin(α/2) on the left of the equation and everything else on the right):
2sin2 (α/2) = 1 − cos α
sin2 (α/2) = (1 − cos α)/2
Solving gives us the following sine of a half-angle identity:
The sign of 
depends on the quadrant
in which α/2 lies.
If α/2 is in the first or second quadrants, the formula uses the positive case:
If α/2 is in the third or fourth quadrants, the formula uses the negative case:
Half Angle Formula - Cosine
Using a similar process, with the same substitution of 
(so 2θ = α) we subsitute into
the identity
cos 2θ = 2cos2 θ − 1 (see cosine of a double angle)
We obtain
Reverse the equation:
Add 1 to both sides:
Divide both sides by 2
Solving for cos(α/2), we obtain:
As before, the sign we need depends on the quadrant.
If α/2 is in the first or fourth quadrants, the formula uses the positive case:
If α/2 is in the second or third quadrants, the formula uses the negative case:
Example 1
Find the value of sin 15° using the sine half-angle relationship given above.
Example 2
Find the value of cos 165° using the cosine half-angle relationship given above.
Example 3:
Show that 
Exercises: Evaluating and Proving Half-Angle Identities
1. Use the half angle formula to evaluate sin 75°.
2. Find the value of 
if 
where 0° < α < 90°.
3. Prove the identity: 
4. Prove the identity: 
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