Half-Angle Identities

To find the trigonometric ratios of half of the standard angles, we use half-angle formulas. In this step-by-step guide, you will learn more about the half-angle formulas.

Tutor-style math help

Half-Angle Identities: what to notice and how to work it

Trigonometry skill

Trigonometry connects an angle to a triangle ratio, a unit-circle coordinate, or a repeating graph. Choosing the right picture makes the problem much easier.

What to notice first

Decide whether the problem is triangle-based, circle-based, or graph-based. Then use the matching definition.

Common student mistake

Do not mix degrees and radians. The angle unit must match the formula, graph scale, or calculator setting.

Key formulas and cues

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)

\(\tan\theta=\frac{\sin\theta}{\cos\theta}\)

\(\sin^2\theta+\cos^2\theta=1\)

A reliable path

Choose the modelUse a right triangle, the unit circle, or a transformed graph.
Track unitsConvert degrees and radians when needed.
Use identitiesReplace complicated trig expressions with equivalent simpler ones.

Worked examples

Right-triangle sine

Example: opposite = 5, hypotenuse = 13

Sine is opposite over hypotenuse.
Substitute 5 and 13.
Leave the ratio simplified.

Answer: \(\sin\theta=\frac5{13}\)

Unit-circle cosine

Example: \(\cos(0)\)

At angle 0, the point is (1, 0).
Cosine is the x-coordinate.
Read the x-value.

Answer: \(1\)

Open pop-up practice

Try one before moving on

Try: In a right triangle, tangent equals which ratio?

Answer: Opposite over adjacent.

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

Step-by-step guide to half-angle identities

The half angle formulas are derived from double angle formulas and are expressed in terms of half angles such as \(\frac{\theta }{2}, \frac{x}{2}, \frac{A}{2}\). Half-angle formulas are used to find the exact values of trigonometric ratios of angles such as \(22.5°\) (which is half of the standard angle of \(45°\)), \(15°\) (which is half of the standard angle of \(30°\)).

Here is a list of important half-angle formulas:

\(\color{blue}{sin\:\left(\frac{A}{2}\right)=\pm \sqrt{\frac{1-\:cos\:A}{2}}}\)
\(\color{blue}{cos\:\left(\frac{A}{2}\right)=\pm \sqrt{\frac{1+\:cos\:A}{2}}}\)
\(\color{blue}{tan\:\left(\frac{A}{2}\right)=\pm \sqrt{\frac{1-\:cos\:A}{1+cos\:A}}=\frac{sin\:A}{1+cos\:A}=\frac{1-cos\:A}{sin\:A}}\)