Sum, Difference and Double-Angle Identities

Sine is not a straight-line function, so it cannot be additive the way multiplication by a constant is. sin(A + B) does not equal sin(A) + sin(B). Instead the angles interact: rotating by A and then by B mixes both the sine and cosine of each piece. The exact rule is sin(A + B) = sin(A)cos(B) + cos(A)sin(B), and the matching one for cosine is cos(A + B) = cos(A)cos(B) - sin(A)sin(B). Swap the plus for a minus and flip the sign on the middle term to get the difference versions: sin(A - B) = sin(A)cos(B) - cos(A)sin(B), and cos(A - B) = cos(A)cos(B) + sin(A)sin(B).

These formulas let you find exact values for angles that are not standard by themselves but can be built from ones you know, like 15, 75, or 105 degrees, using combinations of 30, 45, and 60. They also explain where the double-angle identities come from: set B equal to A in the sum formula and the cross terms collapse. sin(2x) = sin(x)cos(x) + cos(x)sin(x) = 2sin(x)cos(x), and cos(2x) = cos²(x) - sin²(x), which can be rewritten using sin²(x) + cos²(x) = 1 as 1 - 2sin²(x) or 2cos²(x) - 1.

These identities sit right next to the Pythagorean identity and the law of sines and cosines in the toolkit for this unit. Any problem that asks for an exact trig value outside the basic angles, or that asks you to simplify an expression with 2x inside it, is pointing you toward one of these formulas. Keep the sum and double-angle versions ready to rewrite on the spot, since no formula sheet is given.