Trigonometric Equations
What is a trigonometric equation? A trigonometric equation is an equation that contains trigonometric functions of an unknown angle. Unlike a trigonometric identity (which holds for every value of the variable), a trigonometric equation is satisfied only by particular values. Those values are called the roots (or solutions) of the equation.
Because sine, cosine, and tangent are periodic, a trigonometric equation that has any solution at all will usually have infinitely many solutions. We therefore distinguish:
- Principal solutions — the solutions in one basic interval, typically $[0, 2\pi)$.
- General solution — a compact formula that captures all solutions by adding integer multiples of the period.
1. The Three Elementary (Simplest) Equations
Every method for solving trigonometric equations ultimately reduces the problem to one or more of these three basic forms.
| Equation | Existence Condition | General Solution | Period of Family |
|---|---|---|---|
| $\sin x = a$ | $|a| \le 1$ | $$x = n\pi + (-1)^{n}\arcsin a,\quad n \in \mathbb{Z}$$ | $2\pi$ |
| $\cos x = a$ | $|a| \le 1$ | $$x = 2n\pi \pm \arccos a,\quad n \in \mathbb{Z}$$ | $2\pi$ |
| $\tan x = a$ | All real $a$ | $$x = n\pi + \arctan a,\quad n \in \mathbb{Z}$$ | $\pi$ |
Why these formulas look the way they do
Sine. On the unit circle, the horizontal line $y = a$ (with $|a|\le 1$) intersects the circle at two points per revolution:
- Family 1 (Quadrant I side): $x = \arcsin a + 2k\pi$
- Family 2 (Quadrant II side): $x = \pi - \arcsin a + 2k\pi$