Half-Angle Formulas
You know cos 30 degrees exactly, but the angle you need is 15 degrees, which is half of it. The half-angle formulas turn a known angle into the sine, cosine, or tangent of half that angle, with no calculator and no new memorized values.
By the end of this page you can:
- State the half-angle formulas for sin(x/2), cos(x/2), and tan(x/2).
- Choose the correct sign by checking the quadrant of x/2.
- Evaluate an exact value such as cos 15 degrees from a known angle.
The floor. sin(x/2) and cos(x/2) each equal plus or minus a square root built from cos x, and the sign is decided by which quadrant x/2 lands in, not by the sign of x.
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart, Redlin, Watson, Precalculus 7e |
| Chapter | 7 |
| Section | 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas |
| Open alternate | OpenStax Precalculus 2e, 9.3 Double-Angle, Half-Angle, and Reduction Formulas |
| Difficulty | core |
| Time | 30 min |
Before You Start
The Idea
Start from the two forms of the cosine double-angle identity: cos 2x equals 1 minus 2 sin squared x, and cos 2x equals 2 cos squared x minus 1. Solve each for the squared term, then replace x by x/2 (so 2x becomes x). Taking a square root introduces a plus or minus sign, and that sign is fixed by the quadrant where x/2 lives, because sine and cosine are positive in some quadrants and negative in others. The tangent half-angle formulas come from dividing the sine result by the cosine result, and they simplify to two sign-free fractions.
Key Formulas
\[ \sin\frac{x}{2} = \pm\sqrt{\frac{1 - \cos x}{2}} \]
\[ \cos\frac{x}{2} = \pm\sqrt{\frac{1 + \cos x}{2}} \]
\[ \tan\frac{x}{2} = \frac{1 - \cos x}{\sin x} = \frac{\sin x}{1 + \cos x} \]
The plus or minus in the sine and cosine formulas is chosen by the quadrant of x/2. The two tangent forms need no sign choice; they already carry the correct sign.
Source: Stewart 7e, Section 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas.
Worked Example
Find the exact value of cos 15 degrees.
Write 15 degrees as half of 30 degrees, so x equals 30 degrees and x/2 equals 15 degrees. Apply the cosine half-angle formula with cos 30 degrees equal to the square root of 3 over 2:
\[ \cos 15^\circ = \pm\sqrt{\frac{1 + \cos 30^\circ}{2}} = \pm\sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \]
Since 15 degrees is in the first quadrant, cosine is positive, so take the plus sign.
Show the answer
Combine the fraction inside the radical:
\[ \cos 15^\circ = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2} \]
As a decimal check, the square root of 3 is about 1.7320508, so 2 plus that is about 3.7320508. Its square root is about 1.9318517, and dividing by 2 gives about 0.9659258, which matches cos 15 degrees on a calculator.
Connections
This builds directly on the double-angle formulas, since each half-angle formula is solved out of a cos 2x form. It also depends on reading quadrant signs for sine and cosine. It leads into solving trigonometric equations by identities and factoring, where rewriting a half-angle expression is often the move that makes an equation solvable. It does not cover the double-angle formulas, which are their own topic.
Resources
- Stewart, Redlin, Watson, Precalculus 7e, Section 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas (primary).
- OpenStax Precalculus 2e, Section 9.3 Double-Angle, Half-Angle, and Reduction Formulas (alternate).
Last updated: 2026-06-24