Trigonometry Calculator

A trigonometric equation is an equation that involves trigonometric functions ($\sin$, $\cos$, $\tan$, etc.) of an unknown angle. The goal is to find all values of the angle that satisfy the equation.

Because trigonometric functions are periodic, most trig equations have infinitely many solutions. We often express solutions in two forms:

1. Principal solutions: Solutions in a specific interval, typically $[0, 2\pi)$ or $[0°, 360°)$
2. General solutions: All solutions, written using $+ 2n\pi$ (or $+ 360°n$) where $n$ is any integer

For example, $\sin x = \frac{1}{2}$ has principal solutions $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$, and general solutions $x = \frac{\pi}{6} + 2n\pi$ and $x = \frac{5\pi}{6} + 2n\pi$.

Key identities used in solving trig equations:

• Pythagorean: $\sin^2 x + \cos^2 x = 1$
• Double angle: $\sin 2x = 2\sin x \cos x$, $\cos 2x = \cos^2 x - \sin^2 x$
• Sum-to-product and product-to-sum formulas

Method 1: Isolation and Inverse Functions

For simple equations, isolate the trig function and apply the inverse:

$\sin x = a \implies x = \arcsin(a) \text{ and } x = \pi - \arcsin(a)$

$\cos x = a \implies x = \pm \arccos(a)$

$\tan x = a \implies x = \arctan(a) + n\pi$

Method 2: Factoring

When the equation can be factored:

$\sin^2 x - \sin x = 0 \implies \sin x(\sin x - 1) = 0$

So $\sin x = 0$ or $\sin x = 1$, giving $x = 0, \pi, \frac{\pi}{2}$ in $[0, 2\pi)$.

Method 3: Using Identities to Simplify

Replace complex expressions using identities:

Example: Solve $\cos 2x = \cos x$

Using $\cos 2x = 2\cos^2 x - 1$:
$2\cos^2 x - 1 = \cos x$
$2\cos^2 x - \cos x - 1 = 0$
$(2\cos x + 1)(\cos x - 1) = 0$

So $\cos x = -\frac{1}{2}$ or $\cos x = 1$.

Method 4: Substitution

For equations with multiple trig functions, substitute $t = \sin x$ or $t = \cos x$:

$2\sin^2 x + 3\cos x - 3 = 0$

Using $\sin^2 x = 1 - \cos^2 x$: $2(1 - \cos^2 x) + 3\cos x - 3 = 0$ → $2\cos^2 x - 3\cos x + 1 = 0$

Method 5: Squaring Both Sides (with checking)

Sometimes useful, but always verify solutions as squaring can introduce extraneous roots.

Summary of Reference Angles

Comparison of Methods

• Forgetting periodic solutions: $\sin x = 0.5$ has two solutions per period, not one. Always consider all quadrants where the function has the given sign.
• Dividing by a trig function: Dividing by $\sin x$ or $\cos x$ can lose solutions where that function equals zero. Factor instead.
• Not checking extraneous solutions: When squaring both sides, always substitute back to verify. Squaring can introduce false solutions.
• Confusing degrees and radians: Ensure consistency. $\sin(30) \neq \sin(30°)$ in most calculators and programming contexts.
• Ignoring domain restrictions: $\sin x = 2$ has no real solutions since $-1 \leq \sin x \leq 1$.