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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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