An identity is true for every angle; an equation like $2\sin x - 1 = 0$ is true only for some. Solving one uses everything from reference angles and inverse trig functions. The new part is finding all the solutions.
Key idea
Isolate the trig function. Find the reference angle from the size of the value. List every angle in one period, $[0, 2\pi)$, in the quadrants where the sign is right. For all solutions, add whole periods: $+2k\pi$ for sine and cosine, $+k\pi$ for tangent.
Why it works
On the unit circle, $\sin x = \frac{1}{2}$ means the point is at height $\frac{1}{2}$. The horizontal line $y = \frac{1}{2}$ crosses the circle at two points, one in Quadrant I and its mirror image in Quadrant II. A calculator's $\sin^{-1}$ can only report one of them, the one in its range. Then every full turn repeats both.
Worked examples
Worked example: Two solutions
Solve $2\sin x - 1 = 0$ on $[0, 2\pi)$, then give all solutions.
- Isolate: $\sin x = \frac{1}{2}$.
- Reference angle $\frac{\pi}{6}$. Sine is positive in Quadrants I and II: $x = \frac{\pi}{6}$ and $x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$.
- All solutions: $x = \frac{\pi}{6} + 2k\pi$ or $x = \frac{5\pi}{6} + 2k\pi$, for any integer $k$.
Answer: $\frac{\pi}{6}, \frac{5\pi}{6}$ in $[0, 2\pi)$
Worked example: A negative value
Solve $2\cos x + \sqrt{3} = 0$ on $[0, 2\pi)$.
- $\cos x = -\frac{\sqrt{3}}{2}$. Reference angle $\frac{\pi}{6}$.
- Cosine is negative in Quadrants II and III: $\pi - \frac{\pi}{6} = \frac{5\pi}{6}$ and $\pi + \frac{\pi}{6} = \frac{7\pi}{6}$.
Answer: $\frac{5\pi}{6}, \frac{7\pi}{6}$
Worked example: Tangent
Solve $\tan x = -1$ on $[0, 2\pi)$.
- Reference angle $\frac{\pi}{4}$; tangent is negative in Quadrants II and IV.
- $x = \frac{3\pi}{4}$ and $x = \frac{7\pi}{4}$, exactly $\pi$ apart, as tangent's period predicts.
Answer: $\frac{3\pi}{4}, \frac{7\pi}{4}$
Common mistake
Stopping at the one answer a calculator gives. $\sin^{-1}\frac{1}{2} = \frac{\pi}{6}$ is only one solution. Draw the horizontal (for sine) or vertical (for cosine) line on the unit circle and count where it crosses: usually twice per turn. Check each answer by substituting it back.