Double-Angle Formulas
A surveyor knows the sine and cosine of an angle of elevation and needs the sine of twice that angle to model a reflected sightline. There is no need to find the angle itself. The double-angle formulas turn the trig values of x directly into the trig values of 2x.
By the end of this page you can:
- Compute sin 2x, cos 2x, and tan 2x from sin x and cos x.
- Choose the right form of cos 2x for a given problem.
- Recognize each double-angle formula as an addition formula with both angles equal.
The floor. Memorize sin 2x = 2 sin x cos x and cos 2x = cos squared x minus sin squared x. Everything else here follows from those two.
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart, Redlin, Watson 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
A double-angle formula is just an addition formula in which both angles are the same. Start from sin(a + b) = sin a cos b + cos a sin b and set b equal to a. The right side collapses to 2 sin a cos a, which is sin 2a. The same move on cos(a + b) and tan(a + b) gives the cosine and tangent forms.
The cosine formula has three equivalent forms because you can replace sin squared x or cos squared x using the Pythagorean identity. Keep all three in reach: the version with only cosine is the one you want when a later step needs cos 2x written in terms of cos x alone.
Key Formulas
\[ \sin 2x = 2 \sin x \cos x \]
\[ \cos 2x = \cos^2 x - \sin^2 x = 1 - 2\sin^2 x = 2\cos^2 x - 1 \]
\[ \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \]
Source: Stewart 7e, Section 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas.
Worked Example
Suppose sin x = 3/5 and x lies in the first quadrant. Find sin 2x and cos 2x.
Step 1. Find cos x. Since sin squared x plus cos squared x equals 1, cos squared x = 1 minus 9/25 = 16/25. In the first quadrant cosine is positive, so cos x = 4/5.
Step 2. Apply the sine formula: sin 2x = 2 sin x cos x.
Step 3. Apply a cosine form. Using cos 2x = 1 minus 2 sin squared x keeps the arithmetic short here.
Show the answer
sin 2x = 2 times (3/5) times (4/5) = 24/25.
cos 2x = 1 minus 2 times (3/5) squared = 1 minus 2 times (9/25) = 1 minus 18/25 = 7/25.
Check with a second form: cos 2x = cos squared x minus sin squared x = 16/25 minus 9/25 = 7/25. The two forms agree.
Connections
This builds on the addition and subtraction formulas, since every double-angle formula is that wider formula with equal angles. It supports the half-angle formulas, which come from solving the cosine double-angle forms for sin squared x and cos squared x. It is also a standard tool when solving trigonometric equations by identities and factoring, where rewriting sin 2x or cos 2x in terms of a single function lets you factor.
Resources
- Stewart, Redlin, Watson 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