Double-angle identities · trigonometric identity
sin 2x = 2 sin x cos x — Double-angle identity for sine
When to use it
The question is in 2x and the answer (or the next step) is in x — or you need to integrate/solve something containing sin x cos x, which is this formula run backwards.
Why it is true
It is the sine sum formula with both angles set equal: sin(x + x) = sin x cos x + cos x sin x = 2 sin x cos x.
The full line-by-line version is on the proof page for double-angle identity for sine; the “how would I find this myself” version is in the derivation.
Where it comes from
Spot the pattern in a product. Going the other way is the useful skill: whenever you see sin x cos x, multiply and divide by 2 and the double-angle form appears. This is how ∫sin x cos x dx is done in one line.
- See sin x cos xTrigger.
- Write it as (1/2)·(2 sin x cos x)Multiply by 2 and divide by 2.
- = (1/2) sin 2xDouble-angle form.
Worked examples
- Derive the double-angle formulas from the sum formulasSet B = A in each sum formula; cosine then yields two further forms via sin² + cos² = 1.
- ∫ sin x cos x dx — two wayssin²x/2 + C = −cos 2x/4 + C (the forms differ by a constant)
Checked at these angles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 0° | 0 | 0 | yes |
| 7.5° | 0.258819 | 0.258819 | yes |
| 15° | 0.5 | 0.5 | yes |
| 18° | 0.587785 | 0.587785 | yes |
| 22.5° | 0.707107 | 0.707107 | yes |
| 30° | 0.866025 | 0.866025 | yes |
| 37° | 0.961262 | 0.961262 | yes |
| 45° | 1 | 1 | yes |
| 53° | 0.961262 | 0.961262 | yes |
| 60° | 0.866025 | 0.866025 | yes |
| 67.5° | 0.707107 | 0.707107 | yes |
| 75° | 0.5 | 0.5 | yes |
| 90° | 0 | 0 | yes |
| 120° | -0.866025 | -0.866025 | yes |
| 135° | -1 | -1 | yes |
| 150° | -0.866025 | -0.866025 | yes |
| 180° | -0 | -0 | yes |
| 210° | 0.866025 | 0.866025 | yes |
| 240° | 0.866025 | 0.866025 | yes |
| 270° | 0 | 0 | yes |
| 300° | -0.866025 | -0.866025 | yes |
| 330° | -0.866025 | -0.866025 | yes |
| 360° | -0 | -0 | yes |
The mistake students make
Writing sin 2x = 2 sin x. The factor 2 multiplies only the angle inside, and the correct expansion has TWO factors: 2 sin x cos x.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for double-angle identities.
Related identities
- cos 2x = cos²x − sin²xDouble-angle identity for cosine (form 1)detailsproof
- tan 2x = 2 tan x / (1 − tan²x)Double-angle identity for tangentdetailsproof
- sin(A + B) = sin A cos B + cos A sin BSine of a sumdetailsproof
- sin³x = (3 sin x − sin 3x)/4Reduction formula for sin³detailsproof
Category hub: Double-angle identities · all identities: /identities