Trig Identities
An identity is an equation that's true for every angle — not one you solve, one you lean on. Here are the two families you'll use constantly, with the machinery running live so you can try to break them. For Grade 12.
sin²θ + cos²θ = 1 — it's just Pythagoras
On the unit circle, sin θ and cos θ are the two legs of a right triangle whose hypotenuse is the radius — length 1. Pythagoras does the rest:
Try to unbalance it 🟢🔵
The addition formulas
What's the sine of a sum of two angles? Not sin a + sin b — sine isn't a length you can just add. The real recipe mixes both functions:
cos(a + b) = cos a cos b − sin a sin b
Check it for every pair 🟣
Slide a and b. The purple number computes the left side directly; the green one builds it from the formula. They never disagree.
Why these two matter
They're the algebra of trig
Every "prove this identity" question is these two plus rearranging. Even the double-angle formulas are just the addition formulas with b = a: sin 2a = 2 sin a cos a.
They power the waves
The addition formula is how a shifted wave sin(x + C) gets rewritten as a mix of sin x and cos x — the trick behind phase shifts in the Radians & Graphs demo and behind all of Fourier.