Lesson 14 — Trigonometric equations and inequalities

Structure of the solutions

Equations with sine

For $\sin x = a$ with $a \in [-1, 1]$: $x = \arcsin a + 2k\pi \quad \text{ou} \quad x = \pi - \arcsin a + 2k\pi, \quad k \in \mathbb{Z}$

Where $\arcsin: [-1, 1] \to [-\pi/2, \pi/2]$ is the principal inverse.

Equations with cosine

For $\cos x = a$ with $a \in [-1, 1]$: $x = \pm \arccos a + 2k\pi, \quad k \in \mathbb{Z}$

Where $\arccos: [-1, 1] \to [0, \pi]$.

Equations with tangent

For $\tan x = a$ with $a \in \mathbb{R}$: $x = \arctan a + k\pi, \quad k \in \mathbb{Z}$

Where $\arctan: \mathbb{R} \to (-\pi/2, \pi/2)$.

General recipe

To solve a trigonometric $f(x) = 0$:

1. Reduce to basic equations $\sin x = a$, $\cos x = a$, $\tan x = a$ via identities.
2. List all solutions in the interval $[0, 2\pi)$ (or $[-\pi, \pi]$).
3. Generalize by adding $2k\pi$ (or $k\pi$ for tangent).