Elementary trigonometric equations

The simplest trigonometric equations have a trigonometric function set equal to a constant. Their solutions are obtained with the inverse functions and, because of periodicity, form one or more infinite families of angles.

In summary — Elementary equations

• $\sin x = k$ (with $|k|\le 1$): $x = \arcsin k + 2k_1\pi$ or $x = \pi - \arcsin k + 2k_1\pi$, with $k_1\in\mathbb{Z}$. If $|k|>1$, no solution.
• $\cos x = k$ (with $|k|\le 1$): $x = \pm \arccos k + 2k_1\pi$, with $k_1\in\mathbb{Z}$. If $|k|>1$, no solution.
• $\tan x = k$: $x = \arctan k + k_1\pi$, with $k_1\in\mathbb{Z}$ (always a single family, period $\pi$).

Example

Solve $\sin x = \dfrac{\sqrt{3}}{2}$.

Since $\arcsin\left(\sqrt{3}/2\right) = \pi/3$, we have $x = \frac{\pi}{3} + 2k\pi \quad\text{or}\quad x = \pi - \frac{\pi}{3} + 2k\pi = \frac{2\pi}{3} + 2k\pi.$

Links

Topics: Equazioni goniometriche
Concepts: Equazione elementare · Equazione goniometrica · Periodicita
Functions: Coseno · Seno · Tangente
Methods: Goniometriche equazioni
Skills: Risolvere equazioni · Usare formule