Solving Trig Equations

An equation like \sin\theta = k has one obvious answer: type \sin^{-1}(k) into a calculator and you get a single angle. But the sine curve repeats and is symmetric, so in the range 0^\circ to 360^\circ there is usually a second solution as well.

For sine, the second solution is 180^\circ - \theta.
For cosine, the second solution is 360^\circ - \theta.
Tangent repeats every 180^\circ, so the next solution is \theta + 180^\circ.

Find the principal value \theta = \sin^{-1}(k) (or \cos^{-1}, \tan^{-1}) with the inverse function.
Use the graph's symmetry to read off the other solutions inside the range.
\sin\theta = k \;\Rightarrow\; \{\,\theta,\; 180^\circ - \theta\,\}
\cos\theta = k \;\Rightarrow\; \{\,\theta,\; 360^\circ - \theta\,\}
\tan\theta = k \;\Rightarrow\; \{\,\theta,\; \theta + 180^\circ\,\}

Two crossings

Here is y = \sin(x^\circ) with the horizontal line y = 0.5. Solving \sin\theta = 0.5 means finding where they cross — and there are two such points in 0^\circ to 360^\circ: at 30^\circ and at 180^\circ - 30^\circ = 150^\circ.