How the Sum and Difference Identities Work
The sum and difference identities express the sine, cosine, and tangent of a combined angle (A + B or A − B) in terms of the sine, cosine, and tangent of the two individual angles A and B. They let you find exact trig values for angles that are not on the standard unit circle (like 15° or 75°) by breaking them into sums or differences of angles you already know (like 45°, 30°, or 60°), and they underlie many identities used throughout trigonometry and calculus.
Formula and method
The four core identities are: sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB (note the sign flips); and tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB). This calculator converts both angles to radians internally if you entered degrees, applies the sine and cosine identities directly, then derives the tangent result as sin(A ± B) / cos(A ± B) so it stays correct even where the individual tangent of A or B would be undefined.
Common sources of error
- Sign confusion in the cosine identity: cos(A + B) uses minus and cos(A − B) uses plus — the opposite of what you might expect from the sine identity.
- Mixing degrees and radians: both A and B must be in the same unit before you add or subtract them; select the correct unit before calculating.
- Assuming tan(A ± B) always exists: it is undefined whenever cos(A ± B) = 0, which happens at 90°, 270°, and every odd multiple of 90° from the combined angle.
Checking your result
A quick sanity check: sin(A ± B) and cos(A ± B) should each stay between −1 and 1 for any real angles, since they equal the sine and cosine of the combined angle A ± B. If your calculated sin(A ± B) matches sin(A ± B) computed directly from the combined angle, the identity was applied correctly — this calculator's "Combined angle" result lets you verify that independently.
Applications
Sum and difference identities are used to derive exact values for non-standard angles, to prove other trigonometric identities (including the double-angle and product-to-sum formulas), and to simplify expressions in calculus, physics (wave interference, phase shifts), and engineering (signal processing, AC circuit analysis).