Equations of the form $\sin n\theta = k$, $\cos n\theta = k$ and $\tan n\theta = p$ are called trigonometric equations in multiples of the angle. The unknown appears as a multiple $n\theta$ rather than as $\theta$ itself, and $n$ is usually a small positive integer such as $2$ or $3$.

The substitution method

The standard approach is to replace the multiple with a single letter:

• Let $u = n\theta$.
• Transform the given interval for $\theta$ into an interval for $u$ by multiplying both endpoints by $n$.
• Solve the simpler equation for $u$ over that new interval.
• Convert each solution back using $\theta = \dfrac{u}{n}$.

Finding all solutions in the interval

A trigonometric equation normally has infinitely many solutions, so the interval matters. Use the symmetry and periodicity of each function:

• For $\sin u = k$, one solution is $u = \sin^{-1} k$; the second in a full turn is $u = 180^\circ - \sin^{-1} k$.
• For $\cos u = k$, the two solutions in a full turn are $u = \cos^{-1} k$ and $u = 360^\circ - \cos^{-1} k$.
• For $\tan u = p$, the tangent repeats every $180^\circ$, so add multiples of $180^\circ$ to $u = \tan^{-1} p$.

Because the interval for $u$ is $n$ times as wide as the interval for $\theta$, it typically contains $n$ times as many solutions — for example, an interval of length $360^\circ$ for $u = 3\theta$ becomes a range of $1080^\circ$, so expect several solutions before dividing back.

Equations mixing two multiples

When an equation contains two different multiples, such as $2\sin 2\theta = \cos 2\theta$, first reduce it to a single trigonometric ratio. Dividing both sides by $\cos 2\theta$ gives $2\tan 2\theta = 1$, which is then solved by the substitution method above. Always check that dividing by a cosine does not discard valid solutions where that cosine is zero.

Checking the answer

After dividing each $u$-value by $n$, confirm that every resulting $\theta$ lies inside the original interval. Discard any that fall outside, and state the final answers to the required degree of accuracy.