Sum and difference identities · trigonometric identity
sin(A + B) = sin A cos B + cos A sin B — Sine of a sum
When to use it
The angle itself is a sum and you need the functions separately — to evaluate exactly (sin 75° = sin(45° + 30°)), to expand, or to recognise the pattern running backwards.
Why it is true
Rotating by A + B is the same as rotating by A and then by B. Reading the vertical coordinate of that composed rotation gives the sine sum formula.
The full line-by-line version is on the proof page for sine of a sum; the “how would I find this myself” version is in the derivation.
Where it comes from
Two right triangles stacked. The classical derivation drops perpendiculars from a point at angle A + B and reads the vertical leg as two pieces.
- Draw angle A + B in standard position, unit hypotenuseSet-up.
- Split the vertical leg with a perpendicular at angle ACreates two right triangles whose angles are A and B.
- lower piece = sin A cos BAdjacent leg of the B-triangle times cos B.
- upper piece = cos A sin BOther leg, read with sin B.
- sin(A + B) = sin A cos B + cos A sin BAdd the two pieces.
Worked examples
- Find the exact value of sin 75°(√6 + √2)/4
- Find the exact value of sin 15°(√6 − √2)/4
- Verify tan A + tan B = sin(A + B)/(cos A cos B)Common denominator, then recognise the sine sum.
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° | 0.156434 | 0.156434 | yes |
| 7.5° | 0.346117 | 0.346117 | yes |
| 15° | 0.522499 | 0.522499 | yes |
| 18° | 0.587785 | 0.587785 | yes |
| 22.5° | 0.678801 | 0.678801 | yes |
| 30° | 0.809017 | 0.809017 | yes |
| 37° | 0.902585 | 0.902585 | yes |
| 45° | 0.97237 | 0.97237 | yes |
| 53° | 0.999657 | 0.999657 | yes |
| 60° | 0.987688 | 0.987688 | yes |
| 67.5° | 0.938191 | 0.938191 | yes |
| 75° | 0.85264 | 0.85264 | yes |
| 90° | 0.587785 | 0.587785 | yes |
| 120° | -0.156434 | -0.156434 | yes |
| 135° | -0.522499 | -0.522499 | yes |
| 150° | -0.809017 | -0.809017 | yes |
| 180° | -0.987688 | -0.987688 | yes |
| 210° | -0.587785 | -0.587785 | yes |
| 240° | 0.156434 | 0.156434 | yes |
| 270° | 0.809017 | 0.809017 | yes |
| 300° | 0.987688 | 0.987688 | yes |
| 330° | 0.587785 | 0.587785 | yes |
| 360° | -0.156434 | -0.156434 | yes |
The mistake students make
Writing sin(A + B) = sin A + sin B. The sine of a sum is not the sum of sines — check it with A = B = 45°: sin 90° = 1 but sin 45° + sin 45° ≈ 1.414.
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 sum and difference identities.
Related identities
- sin(A − B) = sin A cos B − cos A sin BSine of a differencedetailsproof
- cos(A + B) = cos A cos B − sin A sin BCosine of a sumdetailsproof
- sin 2x = 2 sin x cos xDouble-angle identity for sinedetailsproof
- tan(A + B) = (tan A + tan B) / (1 − tan A tan B)Tangent of a sumdetailsproof
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