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Half Angle Identities

Rule

Half-Angle Identities

The half-angle identities are special cases of angle difference identities. To evaluate trigonometric functions of half an angle, the following identities can be applied. The sign of each formula is determined by the quadrant in which lies.

These identities are useful to find the exact value of the sine, cosine, or tangent at a given angle. For example, knowing that the exact value of can be found by using the second formula. In this case, the first step is to rewrite as Above, the positive sign was chosen because lies in the first quadrant and cosine is positive there. Below, a proof of the second identity is shown. The other two identities can be proven by following a similar reasoning.

Proof

Proof
First, write the Double-Angle Identity for the cosine. Next, solve this equation for
Solve for
Finally, substitute for to obtain the required identity.