Numbers & Functions · 6 Trigonometric Functions

6.9 A toolkit of identities

Symmetry identities — walking backwards (negative angle) gives the same across-position but opposite height: cos(−θ) = cos θ (cosine is even) and sin(−θ) = −sin θ (sine is odd).

The addition formulas — the workhorses, giving sine and cosine of a sum: sin(α + β) = sin α cos β + cos α sin β cos(α + β) = cos α cos β −sin α sin β Most books simply hand these to you. We’re going to prove them, the honest geometric way — because in Chapter 8 they’ll fall out of a single magical line, but that one-line version secretly uses these formulas, so we need them proven independently first or the Chapter 8 argument would chase its own tail. ♦Why it’s true — skip on a first read, but the real brain-training We’ll prove the sine version. The whole proof is one carefully drawn picture and then reading lengths off it.

Stack two angles: open up β from the horizontal, then a further α on top, so the outermost ray sits at total angle α + β. Mark P at distance 1 along it. Drop a perpendicular from P to the middle ray, foot Q; then from Q, a perpendicular to the horizontal axis, foot R. What we want, sin(α + β), is just the height of P.

P 1 Q R S cos α sin β sin α cos β β α The height of P splits into two stacked pieces. In right triangle OQP: OP = 1 and the angle at O is α, so OQ = cos α and QP = sin α. Piece 1 — height of Q: in triangle OQR, the height is OQ sin β = cos α sin β. Piece 2 — extra height of P above Q: the segment QP is tilted at angle β from the vertical, so its vertical contribution is QP cos β = sin α cos β. Add them: sin(α + β) = cos α sin β | {z } piece 1 + sin α cos β | {z } piece 2 .✓ The cosine version comes out the same way by reading the horizontal pieces, landing on cos(α+β) = cos α cos β −sin α sin β — the minus sign appearing because the horizontal pieces work against each other.

The double-angle formulas — just the addition formulas with α = β: sin(2θ) = 2 sin θ cos θ cos(2θ) = cos2 θ −sin2 θ.

You don’t memorize these separately — you derive them in one line. One family, parent and child.

The honest message: trigonometry is not a pile of thirty formulas. It’s one picture (the walker), one identity (Pythagoras on the circle), and a couple of addition formulas.

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