Solving trigonometric equations and general solutions

Radians for solving

• Level 3 solving usually works in radians: $180° = \pi$ radians, and a full turn is $2\pi$.
• Know the special values: $\sin\dfrac{\pi}{6} = \dfrac{1}{2}$, $\cos\dfrac{\pi}{3} = \dfrac{1}{2}$, $\tan\dfrac{\pi}{4} = 1$.
• Read the domain carefully, e.g. $0 \le x \le 2\pi$ — your answers must lie inside it.

Solving over a given domain

• Isolate the trig function so it reads $\sin x = k$ (or $\cos$, $\tan$).
• Find the principal value with the inverse function.
• Use symmetry to find every solution in the domain:
  • $\sin$: a second solution at $\pi - x$.
  • $\cos$: a second solution at $2\pi - x$ (that is, $\pm x$).
  • $\tan$: solutions repeat every $\pi$.

General solutions

• To capture every solution (no domain limit), add whole periods:

$\sin x = k: \quad x = \sin^{-1}k + 2n\pi \;\text{ or }\; x = \pi - \sin^{-1}k + 2n\pi$

$\cos x = k: \quad x = \pm\cos^{-1}k + 2n\pi$

$\tan x = k: \quad x = \tan^{-1}k + n\pi$

• Here $n$ is any integer ($n = 0, \pm 1, \pm 2, \ldots$).

Equations that need an identity

• If the equation mixes a double angle with single angles, substitute a double angle formula first.
• Aim for a single trig function, then treat the equation as a quadratic and factorise.