Trigonometric Sum and Difference Identities

Sum and difference identities

The sine and cosine of two angles determine the values for their sum and difference. The sine of a sum adds the cross products. The cosine of a sum subtracts the product of the sines from the product of the cosines. Replacing the second angle by its negative gives the difference formulas.

Example

Angles of 30° and 60° add to 90°, whose sine is 1 and cosine is 0. Their difference is minus 30°, whose sine is minus 0.5. Entering 45° for both angles demonstrates the double angle case.

Scope

This tool applies four standard identities and shows the numerical substitutions. It does not process arbitrary symbolic expressions or treat agreement at one point as a general proof. Each input has its own angle unit selector.

Checking the signs

Sine and cosine sums use different signs. The sine sum adds cross products, while the cosine sum subtracts the product of the sines. For a difference, the second sine term is subtracted and the cosine products are added. Check the sign against the quadrant of the combined angle.

For example, enter 30° twice. On the unit circle, sin 30° = 0.5 and cos 30° = √3/2. The sine of the sum is 2 × 0.5 × √3/2 = √3/2 ≈ 0.866025. The cosine of the sum is 3/4 − 1/4 = 0.5. The difference angle is zero, so its sine is 0 and cosine is 1. These inputs demonstrate both the double angle formula and a zero angle check.

Swapping the angles leaves the sum outputs unchanged. The sine of the difference changes sign, while its cosine stays the same. These properties provide useful checks. Outputs are dimensionless ratios, not lengths or angles.

One angle may be entered in degrees and the other in radians. Verify each selector separately. Rounding differences can appear relatively large near zero. Agreement for a numerical example is not proof for all angles. The tool shows application at the supplied angles rather than a symbolic proof.