Revision sheet

The Pythagorean identity

• $\sin^2 x + \cos^2 x = 1$ for every $x$.
• From one value: square, subtract from 1, take the root, choose the sign by quadrant.

Watch out: the identity needs the squares.

Solving sin x = c, cos x = c, tan x = c

1. Reference angle from $|c|$.
2. The two quadrants with the right sign.
3. Write both solutions in $[0, 2\pi)$.

Watch out: there are usually two solutions, not one.

Solving by factoring

1. One side 0, then factor (or set $u = \sin x$).
2. Each factor = 0.
3. Solve each $\sin x = u$ in $[0, 2\pi)$.

Watch out: never divide by $\sin x$ or $\cos x$.

Multiple angles

1. Let $\theta = nx$: it runs over $[0, 2n\pi)$.
2. List every $\theta$ there.
3. Divide each by $n$.

Watch out: $\sin 2x$ has twice as many solutions as $\sin x$.

Sum and difference identities

• $\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$.
• $\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$.

Watch out: $\cos(A + B) \ne \cos A + \cos B$; cosine flips the sign.

Double-angle identities

• $\sin 2x = 2\sin x\cos x$.
• $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$.

Watch out: $\sin 2x \ne 2\sin x$ (a sine is never above 1, but $2\sin x$ can be).

Equations with identities

1. Rewrite with one angle (double-angle identities).
2. One side 0, factor (or $u = \sin x$, $u = \cos x$).
3. Solve each factor in $[0, 2\pi)$.

Watch out: factor, never divide by $\sin x$ or $\cos x$.