Write general solutions to trigonometric equations using the period and the symmetries of $\sin$, $\cos$ and $\tan$

What this dot point is asking

NESA wants you to write down every solution (over all real numbers) to a trigonometric equation, then restrict to a given interval if asked. The general solution captures the periodic and reflective structure of $\sin$, $\cos$ and $\tan$.

The answer

General solution for $\sin \theta = k$ (with $-1 \le k \le 1$)

$\sin$ has period $2 \pi$ and is symmetric about $\theta = \frac{\pi}{2}$: $\sin(\pi - \theta) = \sin \theta$. So solutions are

where $\alpha = \arcsin k$ is the principal value.

A compact way to write the same set: $\theta = n \pi + (-1)^n \alpha$ for $n \in \mathbb{Z}$.

General solution for $\cos \theta = k$ (with $-1 \le k \le 1$)

$\cos$ has period $2 \pi$ and is even: $\cos(-\theta) = \cos \theta$. So solutions are

where $\alpha = \arccos k$ is the principal value.

General solution for IMATH_23

$\tan$ has period $\pi$ (not $2 \pi$). Solutions are

where $\alpha = \arctan k$ is the principal value.

Equations with composite arguments

For $\sin(2 x + 1) = \frac{1}{2}$, treat $u = 2 x + 1$ as the variable, find the general solution for $u$, then back-solve for $x$.

For $\cos 3 x = -\frac{\sqrt{3}}{2}$: $3 x = \pm \frac{5 \pi}{6} + 2 n \pi$, so $x = \pm \frac{5 \pi}{18} + \frac{2 n \pi}{3}$.

The period of $\sin k x$ or $\cos k x$ is $\frac{2 \pi}{|k|}$; the period of $\tan k x$ is $\frac{\pi}{|k|}$.

Restriction to an interval

After writing the general solution, list the values of $n$ that put $\theta$ in the required interval. Use a number line or trial substitution.

Multiple-step equations

Combine general-solution skills with identities:

• IMATH_42 : take both square roots, then solve $\sin x = \pm \sqrt{c}$.
• IMATH_44 : convert to $R \sin(x + \alpha) = c$ first (see auxiliary-angle method).
• IMATH_46 : rearrange to $2 \sin x \cos x - \sin x = 0$, factor as $\sin x (2 \cos x - 1) = 0$, solve each factor.