# Trigonometric Equations - General Solution & Examples

## TL;DR

A trigonometric equation is solved by finding every angle that satisfies it, written as a general solution like x=nπ+(−1)n,α. This article gives the general-solution formulas for sin, cos, and tan, the difference between principal and general solutions, six worked examples, and the sign and periodicity errors that trip students up.

## What Is a Trigonometric Equation?

A **trigonometric equation** is an equation that contains one or more trigonometric ratios of an unknown angle, such as sin x=12, 2cos 2x−1=0, or tan x=3. Solving it means finding every value of the angle that makes the equation true.

Two kinds of answers exist, and keeping them apart is most of the battle:

- **Principal solution.** The solution(s) that lie in the base interval 
0,2π (or 0°,360°). There are usually one or two.

- **General solution.** Every solution, written in terms of an integer n so the periodic copies are all included.

### The General-Solution Formulas

These three results are the spine of the whole topic. In each, n is any integer (n∈Z).

sin x=sin α;⇒;x=nπ+(−1)n,α

cos x=cos α;⇒;x=2nπ±α

tan x=tan α;⇒;x=nπ+α

**Why do the three differ?** It comes down to where each function repeats its value.

**Sine** repeats every 2π but is also symmetric about x=π2, so the (−1)n flips between the two solutions per cycle.

**Cosine** is symmetric about the x-axis, so each value comes from a +α and a −α branch, hence the ±.

**Tangent** has the shortest period, π, so its solutions are spaced a single π apart with no sign flip.

| Equation       | General solution               | Period feeding it |
|----------------|--------------------------------|-------------------|
| sin x=sin α   | x=nπ+(−1)nα                   | 2π                |
| cos x=cos α   | x=2nπ±α                       | 2π                |
| tan x=tan α   | x=nπ+α                         | π                 |

When the equation hands you a value that is not a standard angle, you reach for the inverse trigonometric functions to name α first, then drop it into the formula above.

## How Do You Solve a Trigonometric Equation Step by Step?

The reliable routine is the same every time:

1. **Isolate the trig ratio** so the equation reads sin x=k (or cos, tan).
2. **Find one known angle α** whose ratio equals k — a special angle if possible, otherwise an inverse-function value.
3. **Apply the matching general-solution formula.**
4. **List principal solutions** by setting n=0,1 if the question asks for the base interval.

## Examples of Trigonometric Equations

### Example 1

**Find the principal solutions of sin x=32.**

sin π/3=32, and sine is also positive in the second quadrant.

So x=π/3 and x=π−π/3=2π/3.

**Final answer:** x=π/3, 2π/3.

### Example 2

**Solve cos x=cos x−sin x by first instinct, then correctly. Take 2sin x=2.**

The tempting move with sin x=12 is to write only x=π/4 and stop.

The rescue is the formula. With sin x=sin π/4:

x=nπ+(−1)nπ/4, n∈Z.

**Final answer:** x=nπ+(−1)nπ/4.

### Example 3

**Find the general solution of cos x=−12.**

cos 2π/3=−12, so α=2π/3.

x=2nπ±2π/3, n∈Z.

**Final answer:** x=2nπ±2π/3.

### Example 4

**Solve tan x=−13.**

A reference value is tan π/6=13, so α=−π/6.

x=nπ−π/6, n∈Z.

**Final answer:** x=nπ−π/6.

### Example 5

**Solve 2cos 2x+cos x−1=0.**

Treat it as a quadratic in cos x. Let c=cos x:

2c²+c−1=0, (2c−1)(c+1)=0, c=1/2 or c=−1.

For cos x=1/2: x=2nπ±π/3.

For cos x=−1: x=(2n+1)π.

**Final answer:** x=2nπ±π/3 or x=(2n+1)π.

### Example 6

**Solve tan 3x=cot(x−50°).**

Using tan A=tan B⇒A=n⋅180°+B:

3x=180°n+140°−x.

x=45°n+35°.

The smallest positive value is at n=0: x=35°.

**Final answer:** x=35°.

## Why Solving These Equations Matters - "The signal hits a target value"

The reason trigonometric equations exist is to answer a question every oscillating system asks: _when does the wave reach this level?_ Sound, light, alternating current, planetary orbits, and seasonal temperature all rise and fall periodically.

## Key Takeaways

- A **trigonometric equation** has infinitely many solutions, captured by a general-solution formula with integer n.

- **Principal solution**: restricted to [0, 2π).

- Never divide by a trig term — factor instead, or you lose solutions.

- Match the formula to the function; the three are not interchangeable.