Half Angle Formula — Sin, Cos, Tan, Proof

TL;DR

The half angle formula expresses sin(θ/2), cos(θ/2), and tan(θ/2) using cos(θ), with a ± sign fixed by the quadrant the half-angle lands in. This article gives all three identities, the double-angle proof, the quadrant sign rule, six worked examples in degrees and radians (including 15°, 22.5°, and π/12), and the mistakes that cost marks.

The half angle formulas — also called half angle identities — give:

• sin(θ/2) from cos(θ),
• cos(θ/2) from cos(θ),
• tan(θ/2) from sin(θ) and cos(θ).

Each one carries a ± that you resolve by asking a single question: which quadrant does θ/2 live in?

What Is the Half Angle Formula?

The half angle formula is a set of three trigonometric identities that express the sine, cosine, and tangent of half an angle, θ/2, in terms of the trig functions of the full angle θ. They are the reverse direction of the double-angle identities.

[ \sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}} ] [ \cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}} ] [ \tan\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} ]

How Is the Half Angle Formula Derived?

The cleanest proof starts from a double-angle identity for cosine and works backwards. The cosine half-angle starts from cos(2x) = 2cos^2(x) - 1. Let x=θ/2, so 2x=θ:

[ \cos\theta = 2\cos^2\frac{\theta}{2} - 1 ]

Solve for cos²(θ/2):

[ \cos^2\frac{\theta}{2} = \frac{1 + \cos\theta}{2} \implies \cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}} ]

For sine and tangent, derive similarly by using other forms of the double angle formulas.

How Do You Choose the ± Sign by Quadrant?

The sign on sin(θ/2) and cos(θ/2) is based on where θ/2 sits in the quadrant.

Examples of the Half Angle Formula

Example 1

Find sin(15°) exactly using the half angle formula.

[ 15° = \frac{30°}{2} \implies \sin(15°) = \sqrt{\frac{1 - \cos(30°)}{2}} = \frac{\sqrt{2 - \sqrt{3}}}{2}. ]

Example 2

Find cos(θ/2) when cos(θ) = 3/5 and θ/2 is in Quadrant II.

[ \cos\frac{\theta}{2} = -\sqrt{\frac{1 + \frac{3}{5}}{2}} = -\frac{2\sqrt{5}}{5}. ]

Example 3

Find tan(22.5°) exactly.

[ 22.5° = \frac{45°}{2} \implies \tan(22.5°) = \frac{1 - \cos(45°)}{\sin(45°)} = \sqrt{2} - 1. ]

Example 4

Find cos(π/8) exactly.

[ cos(\frac{\pi}{8}) = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}; ]

Example 5

Given sin(θ) = 4/5, find sin(θ/2).

[ \cos(θ) = \sqrt{1 - \left(\frac{4}{5}\right)^2} = \frac{3}{5}. ]

Example 6

Prove the identity ( \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta} ).

Both sides simplify to sin²(θ).

Conclusion

• The half angle formula provides essential trigonometric identities useful across various mathematical applications in science, engineering, and beyond.