Half-Angle Identities

For any angle or value \(A\), the following relationships are always true.

\[\sin\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1-\cos(A)}{2}}\]

\[\cos\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1+\cos(A)}{2}}\]

\[\tan\left(\frac{A}{2}\right) = \frac{1-\cos(A)}{\sin(A)} = \frac{\sin(A)}{1+\cos(A)}\]

For the half-angle identites of sine and cosine, the sign of the square root is determined by the quadrant in which \(\frac{A}{2}\) is located. The half-angle identity for tangent has two forms, which you can use either depending on which would be more relevant or helpful in a given problem.

Simplifying Expressions

Remember that as we look at more identities, the process is still the same. We just have more identities to potentially use. This means it can be more challenging to decide which identity should be used in a given problem. Try to look for similarities between the given equation or expression and the identities available. And again, just try something and see what happens.

Power Reducing Identities

Another set of identities that are related to the Half-Angle Identities is the Power-Reducing Identities. We get these new formulas by basically squaring both sides of the sine and cosine half-angle formulas, and then the tangent formula is just sine divided by cosine.

\[\sin^2(A) = \frac{1-\cos(2A)}{2}\]

\[\cos^2(A) = \frac{1+\cos(2A)}{2}\]

\[\tan^2(A) = \frac{1-\cos(2A)}{1+\cos(2A)}\]

Use the above formulas to reduce the power on a trig function from being quadratic (power of 2) to being linear (power of 1).

Let's look at an example using a power reducing identity.

Trigonometric Identities