Sum, Difference, and Double-Angle

Sine and cosine are not linear, so sin(a+b) does not equal sin(a) + sin(b). You can check this with easy numbers: sin(30+60) = sin(90) = 1, but sin(30)+sin(60) is about 1.37. The actual identities come from rotating a point on the unit circle by two angles in a row. Starting at angle a and rotating further by b, the new coordinates mix both angles together through products, not sums. Working through that rotation with coordinate geometry gives cos(a+b) = cos(a)cos(b) - sin(a)sin(b), and a similar argument gives sin(a+b) = sin(a)cos(b) + cos(a)sin(b).

Swapping b for negative b flips the sign on any sine of b, since cosine is even and sine is odd. That turns the sum identity into the difference identity: cos(a-b) = cos(a)cos(b) + sin(a)sin(b). Setting b equal to a in the sine sum identity gives the double-angle formula sin(2theta) = 2 sin(theta)cos(theta), one of the most used shortcuts in the unit. A parallel double-angle identity exists for cosine, built the same way from cos(a+b).

These identities are the algebraic engine behind the unwrapped wave picture of this unit: they explain why graphs of sine and cosine shift, stretch, and relate to each other the way they do. They also set up later work solving trigonometric equations and rewriting expressions, where breaking an unfamiliar angle into a sum of familiar ones is the standard first move.