5.3  Half-Angle Formulas

At times is it important to know the value of the trigonometric functions for half-angles. For example, using these formulas we can transform an expression with exponents to one without exponents, but whose angles are multiples of the original angle.

We obtain half-angle formulas from double angle formulas.

Both sin (2A) and cos (2A) are derived from the double angle formula for the cosine:

cos (2A) = cos2(A) − sin2(A) = cos2(A) − (1 − cos2A) = 2cos2(A) − 1. So, . If we now replace A by (1/2)A, and take the square root we get: .

Similarly, we compute the sine half-angle:
cos (2A) = cos2(A) − sin2(A) = 1 − sin2(A) − sin2(A) = 1 − 2sin2(A). So, . If we now replace A by (1/2)A, and take the square root, we get: .

For the tangent of the half-angle, 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 half-angle formulas:

Half-Angle Identities
Sine
Cosine
Tangent

Example: Given that sin(A)= 3/5 and 90o < A < 180o, find sin(A/2).
Solution: First, notice that the formula for the sine of the half-angle 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 sin2(A) + cos2(A) = 1. In this case, we find:
cos2(A) = 1 − sin2(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 90o and 180o, its sine is positive and its cosine is negative. So in this case, cos A = −4/5. We now apply this to the half-angle formula for sine:

. The question is now: is the answer positive or negative? Again we look at the quadrant; since A is between 90o and 180o,  A/2 is between 45o and 90o. 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 half-angle there are additional identities; they have the advantage that the sign is decisive since there is no square root involved. They are:

Tangent Half Angles

In the exercise that follows, you are asked to evaluate the sine or cosine of a half-angle 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 Half-Angle

In the exercise that follows, you are asked to evaluate the tangent of a half-angle 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 Half-Angle