Half Number Identities

High School

Definition

Also known as half angle identities. They are identities that uses sine, cosine, and tangent for half a given angle in trigonometry. They help you to solve trig functions of angles that aren't on a unit circle by using one that is. You won't be able to get all the angles on a unit circle, but it does help you get a closer answer than without half angle identities.

Worked examples

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

The half-angle identity for sine lets you find sine of half an angle using the cosine of the full angle.

\(\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}\)

The half-angle identity for cosine uses the cosine of the full angle; pick the sign based on the quadrant.

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

The tangent half-angle identity has two forms; both avoid the ± ambiguity of sine and cosine versions.

Common mistakes

\(\sin\left(\frac{\theta}{2}\right) = \frac{\sin\theta}{2}\) → \(\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}\) You cannot just divide the function value by 2; you must use the half-angle formula.
Always taking the positive square root in \(\pm\sqrt{\frac{1 - \cos\theta}{2}}\) → Choose ± based on which quadrant \(\frac{\theta}{2}\) lands in The sign depends on whether sine or cosine is positive or negative in that quadrant.
\(\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos\theta}{2}}\) → \(\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}\) Cosine half-angle uses 1 + cos, not 1 − cos; sine uses 1 − cos.

Where you'll use it next

You'll use half-angle identities to integrate powers of trig functions in calculus, solve trig equations that involve half angles, and derive exact values for angles like 15° or 22.5° in advanced trigonometry.

Found in 1 StudyPug lesson

Double-angle identities

Grade 12 Math

In this lesson, we will learn how to make use of the double-angle identities, a.k.a. double-angle formulas to find the sine and cosine of a double angle. It's hard to simplify complex trigonometric functions without these formulas.

12th Grade12th