Sum & Difference Formulas Calculator

Apply sum and difference identities to compute sin(A±B), cos(A±B), and tan(A±B) exactly from known values of A and B. Enter standard angles like 30°, 45°, 60°, 90° and get exact forms like √3/2, −1/2, √3 — with full step-by-step substitution.

Live Calculator · Exact Values · Step-by-Step · Precalculus

The Six Sum & Difference Formulas

sin(A + B)

= sin A cos B + cos A sin B

sin(A − B)

= sin A cos B − cos A sin B

cos(A + B)

= cos A cos B − sin A sin B

cos(A − B)

= cos A cos B + sin A sin B

tan(A + B)

= (tan A + tan B) / (1 − tan A·tan B)

tan(A − B)

= (tan A − tan B) / (1 + tan A·tan B)

The sign pattern for cos is opposite the operation: cos(A+B) uses minus, cos(A−B) uses plus. For sin and tan, the sign matches the operation.

Finding Exact Values for Non-Standard Angles

Standard angles (0°, 30°, 45°, 60°, 90°, …) have exact trig values. But what about 75° or 15°? Decompose them as sums or differences of standard angles:

75° = 45° + 30°
15° = 45° − 30°
105° = 60° + 45°
π/12 = π/4 − π/6 (= 15°)
5π/12 = π/4 + π/6 (= 75°)

Once decomposed, substitute the known exact values and simplify. Rationalize denominators as needed to get the simplest exact form.

Key exact values to memorize: sin 30° = 1/2, cos 30° = √3/2, sin 45° = cos 45° = √2/2, sin 60° = √3/2, cos 60° = 1/2, tan 45° = 1, tan 30° = 1/√3, tan 60° = √3.

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