Sum and Difference Formulas Calculator
Calculate sin, cos, and tan of the sum and difference of two angles.
Enter angles A and B to get sin(A+B), sin(A-B), cos(A+B), cos(A-B), and more.
The angle addition and subtraction formulas let you compute trig values for sums or differences of angles without a calculator — useful for exact values and for simplifying expressions.
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)
One pattern to notice: for sine, the sign in the formula matches the sign in the argument. For cosine, the signs are reversed. That reversal trips up students more than anything else.
A classic use: sin(75°) = sin(45° + 30°) = sin45 cos30 + cos45 sin30 = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4. That is an exact value you would not get from the standard 30-60-90 table alone.
These formulas are provable from the unit circle using rotation matrices. If you rotate a point by A, then by B, the combined rotation is A + B — and expanding that matrix product gives exactly the formulas above.
Tan(A+B) is undefined when the denominator equals zero, i.e., when tan A × tan B = 1. The calculator shows “undefined” in that case.