We obtain halfangle formulas from double angle formulas.
Both sin (2A) and cos (2A) are derived from the double angle formula for the cosine:
cos (2A) = cos^{2}(A) − sin^{2}(A) = cos^{2}(A) − (1 − cos^{2}A) = 2cos^{2}(A) − 1. So, . If we now replace A by (1/2)A, and take the square root we get: .
Similarly, we compute the sine halfangle:
cos (2A) = cos^{2}(A) − sin^{2}(A) =
1 − sin^{2}(A) − sin^{2}(A) = 1 −
2sin^{2}(A).
So,
.
If we now replace A by (1/2)A, and take the square root,
we get: .
For the tangent of the halfangle, tan
(2A),
we combine the identities for sine and cosine:
.
Again replacing A by (1/2)A, we get:
.
The following is a summary of the halfangle formulas:
HalfAngle Identities  






Example:  Given that sin(A)= 3/5 and 90^{o} < A < 180^{o}, find sin(A/2). 
Solution:  First, notice that the formula for the sine of the halfangle involves
not sine, but cosine of the full angle. So we must first find the value of
cos(A). To do this we use the
Pythagorean
identity sin^{2}(A) + cos^{2}(A) = 1. In
this case, we find:
cos^{2}(A) = 1 − sin^{2}(A) = 1 − (3/5)^{2} = 1 − (9/25) = 16/25. The cosine itself will be plus or minus the square root of 16/25. Which is it? Well, to answer that we need to know in which quadrant A is. Since the angle A is between 90^{o} and 180^{o}, its sine is positive and its cosine is negative. So in this case, cos A = −4/5. We now apply this to the halfangle formula for sine: . The question is now: is the answer positive or negative? Again we look at the quadrant; since A is between 90^{o} and 180^{o}, A/2 is between 45^{o} and 90^{o}. That is, it is in the first quadrant  so its sine is positive. So our answer is: sin (A/2) = . 
For the tangent of a halfangle there are additional identities; they have the advantage that the sign is decisive since there is no square root involved. They are:




In the exercise that follows, you are asked to evaluate the sine or cosine of a halfangle given some information about the angle. Some of the answers in this exercise involve a radical under a radical (sometimes called a nested radical). Because of the difficulty in inputting such expressions, you are only to compare the answer you work out with that of the computer.
Sine and Cosine HalfAngle 

In the exercise that follows, you are asked to evaluate the tangent of a halfangle given some information about the angle. Again, some of the answers in this exercise involve a radical under a radical and so you are only to compare the answer you work out with that of the computer.
Tangent HalfAngle 
