When solving trigonometric equations, the argument of the function must be measured in radians if you intend to use the standard periodicity results. The key periods are:

• $\sin\theta$ and $\cos\theta$ repeat every $2\pi$ radians.
• $\tan\theta$ repeats every $\pi$ radians.

Because the solution set is infinite, equations are usually solved over a stated interval, such as $0 \le \theta < 2\pi$ or $0 \le x < \pi$.

Simple equations

A simple equation has the form $\sin\theta = k$, $\cos\theta = k$, or $\tan\theta = k$.

1. Find the principal value using the inverse function, for example $\theta_0 = \arcsin k$. Calculators return values in a restricted range, so this is only one solution.
2. Use the symmetry of the graph to find the other solutions in the interval.

Useful patterns for $0 \le \theta < 2\pi$:

• For $\sin\theta = k$: solutions are $\theta_0$ and $\pi - \theta_0$.
• For $\cos\theta = k$: solutions are $\theta_0$ and $2\pi - \theta_0$ (or $-\theta_0$).
• For $\tan\theta = k$: solutions are $\theta_0$ and $\theta_0 + \pi$.

If the interval is wider, keep adding or subtracting the period until all solutions are listed.

Equations in multiples of the angle

When the equation involves $\sin(n\theta)$, $\cos(n\theta)$, or $\tan(n\theta)$, the argument $n\theta$ runs through its cycle $n$ times faster.

• Solve first for the whole argument $u = n\theta$.
• Adjust the interval: if $0 \le \theta < 2\pi$, then $0 \le u < 2n\pi$.
• Find all solutions for $u$ in that expanded interval.
• Divide each solution by $n$ to recover $\theta$.

This method also works for arguments such as $\theta + \frac{\pi}{3}$ or $2\theta - \frac{\pi}{4}$: treat the entire bracket as a single variable and shift the interval accordingly.

Equations requiring identities

Many equations mix functions or powers, so they must be rewritten before solving.

• Pythagorean identity: replace $\sin^2\theta$ with $1 - \cos^2\theta$, or $\cos^2\theta$ with $1 - \sin^2\theta$, to obtain a quadratic in one function.
• Double-angle identities: use $\sin 2\theta = 2\sin\theta\cos\theta$ and $\cos 2\theta = \cos^2\theta - \sin^2\theta$ to simplify mixed products.
• Factorising: expressions such as $\sin\theta\cos\theta - \sin\theta = 0$ factor as $\sin\theta(\cos\theta - 1) = 0$, giving separate simple equations.
• Tangent form: dividing by $\cos\theta$ converts an equation into one involving $\tan\theta$, provided $\cos\theta \ne 0$.

After rewriting, solve the resulting simple or multiple-angle equation using the methods above.

Checking solutions

• Verify each answer lies inside the required interval.
• Substitute back into the original equation to catch extraneous solutions, especially after squaring or dividing.
• Remember that inverse trigonometric functions return only one value, so always search for the full set of solutions using periodicity and symmetry.