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 where the angle lies.

Signs of trigonometric ratios

These identities are useful when finding the exact value of the sine, cosine, or tangent at a given angle.

Proof

Half-Angle Identities
First, write two of the Double-Angle Identities for cosine.

Sine Identity

Start by solving the first identity written above for
Solve for
Next, substitute for to obtain the half-angle identity for the sine.

Cosine Identity

Start by solving the second identity written at the beginning for
Solve for
Next, substitute for to obtain the half-angle identity for the cosine.

Tangent Identity

To derive the tangent identity, start by recalling the definition of the tangent ratio.
Next, substitute for
Finally, substitute the half-identities for the sine and cosine into the equation above and simplify the right-hand side.
Simplify right-hand side

Extra

Calculating

Consider the calculation of the exact value of

  1. To be able to use the half-angle identities, the angle needs to be rewritten as a certain angle divided by Therefore, rewrite as
  2. Based on the trigonometric ratios of common angles, it is known that
  3. According to the diagram of the quadrants, an angle that measures is in the first quadrant. Therefore, the cosine ratio is positive.
With these three steps and the second identity in mind, the value of can be found.

Simplify
As already mentioned, the positive sign was chosen because lies in the first quadrant where cosine is positive.