Sum And Difference Formula Calculator

Work out sum and difference formula instantly with clear inputs, formula shown and shareable results.

About Sum And Difference Formula Calculator

The compound angle formulas expand a sum or difference into products of single-angle values: sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B. Note the sign flip in the cosine version.

Formulas

Sine

sin(A ± B) = sin A cos B ± cos A sin B

Cosine

cos(A ± B) = cos A cos B ∓ sin A sin B

Worked example

With the default values below, Sum And Difference Formula Calculator returns sin(A + B) of 0.96592583. Change any input above and every figure here is recalculated in your browser.

Inputs used

Angle A: 45 °
Angle B: 30 °

Results

sin(A + B): 0.96592583
cos(A + B): 0.25881905
sin(A − B): 0.25881905
cos(A − B): 0.96592583

Step by step

1Values used
2Sine
3Cosine
4sin(A + B)= 0.96592583
5cos(A + B)= 0.25881905
6sin(A − B)= 0.25881905
7cos(A − B)= 0.96592583

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

A = 45°, B = 30°?

sin 75° ≈ 0.9659258 and cos 15° ≈ 0.9659258 — the same value, as sin 75° = cos 15°.

Why is the cosine sign reversed?

It follows from the rotation matrix product; geometrically, adding angles reduces the cosine while it increases the sine near zero.