Trigonometric Equations

The inverse functions on your calculator return one $\text{principal value}$: ${\sin }^{-1}$ and ${\tan }^{-1}$ give values in $[-90^{\circ },90^{\circ }]$, and ${\cos }^{-1}$ gives values in $[0^{\circ },180^{\circ }]$.

The second solution in each $360^{\circ }$ cycle comes from symmetry: if $\sin \theta =k$ has solution $\alpha$, the other is $180^{\circ }-\alpha$; for $\cos \theta =k$ the other is $360^{\circ }-\alpha$ (equivalently $-\alpha$); for $\tan \theta =k$ solutions repeat every $180^{\circ }$.

Then add or subtract whole periods ($360^{\circ }$ for sine and cosine, $180^{\circ }$ for tangent) until you have every solution in the interval.

For $\sin (2\theta )=k$ with $0^{\circ }\le \theta <360^{\circ }$, substitute $x=2\theta$. The interval for $x$ becomes $0^{\circ }\le x<720^{\circ }$ — two full cycles, so up to four solutions.

Find all $x$ in the new interval, then convert back: $\theta =\frac{x}{2}$.

For a shifted angle like $\theta -40^{\circ }$, shift the interval by the same amount before solving.

Equations like $2{\sin }^2\theta -\sin \theta -1=0$ are quadratics in $\sin \theta$. Factorise (or use the formula) treating $\sin \theta$ as the unknown.

Each root gives a separate trigonometric equation. Reject any root outside the range of the function — for instance $\sin \theta =2$ has no solutions.

Never divide both sides by $\sin \theta$ or $\cos \theta$: you lose the solutions where that function is zero. Factorise instead.

When an equation mixes functions, use an identity to rewrite it in terms of one. The usual choices: replace ${\cos }^2\theta$ with $1-{\sin }^2\theta$ (or vice versa); replace $\frac{\sin \theta }{\cos \theta }$ with $\tan \theta$; replace ${\sec }^2\theta$ with $1+{\tan }^2\theta$; expand $\sin 2\theta$ or $\cos 2\theta$.

Equations like $a\sin \theta +b\cos \theta =c$ are solved by first writing the left side in harmonic form $R\sin (\theta +\alpha )$.