# Trigonometric equations and inequalities

> AP Precalculus · Trigonometric and Polar Functions
> Source: https://www.owlsprep.com/study/ap-precalculus-u3-trigonometric-equations-and-inequalities/

This module covers solving linear and quadratic trigonometric equations, and trigonometric inequalities on restricted and general domains. We use inverse trigonometry, unit circle symmetry, and periodicity to find all valid solutions, with AP-style worked examples.

**Prerequisites:** Unit circle definitions of sine, cosine, tangent; [Pythagorean and reciprocal trigonometric identities](https://www.owlsprep.com/study/ap-precalculus-u3-trigonometric-identities/); Evaluating inverse trigonometric functions

## Learning objectives

- Solve linear trigonometric equations on restricted and general domains
- Solve quadratic trigonometric equations using identities and factoring
- Solve trigonometric inequalities by identifying critical points and testing intervals
- Apply symmetry and periodicity to find all solutions correctly

## Overview of Trigonometric Equations and Inequalities

Trigonometric equations and inequalities are statements involving trigonometric functions that require finding all input values that satisfy the given relationship. Unlike polynomial or rational equations, trigonometric functions are periodic, meaning most non-restricted problems have infinitely many solutions.

On the AP Precalculus exam, questions almost always specify a restricted domain (most commonly $0 \leq x < 2\pi$), but may also ask for a general solution or to count the number of solutions in a given interval. This topic accounts for ~12% of Unit 3 and appears in both multiple-choice and free-response sections.

**Periodicity of Solutions** — All solutions of a trigonometric equation can be written as a base solution plus a multiple of the function's period, since the function repeats its values every full period.

## Solving Linear Trigonometric Equations

**Linear Trigonometric Equation** — An equation where the trigonometric function is raised only to the first power, where $a,b,c,d$ are real constants.

*Notation:* $af(bx + c) = d$

The core solving strategy follows four key steps: 1. Isolate the trigonometric term on one side of the equation; 2. Check if the resulting right-hand side is within the range of the trigonometric function; 3. Find reference solutions for the argument using the unit circle or inverse trigonometry; 4. Use symmetry to find all solutions for the argument, solve for $x$, then filter to the required domain.

**Worked example:** Find all solutions to $2\sin\left(3x - \frac{\pi}{6}\right) = 1$ on the domain $0 \leq x < 2\pi$.

1. 1. Isolate the trigonometric term by dividing both sides by 2:
2. $$\sin\left(3x - \frac{\pi}{6}\right) = \frac{1}{2}$$
3. $\frac{1}{2}$ is between $-1$ and $1$, so real solutions exist. Let $\theta = 3x - \frac{\pi}{6}$.
4. 2. Find all solutions for $\theta$ using unit circle symmetry:
5. $$\theta = \frac{\pi}{6} + 2\pi n, \quad \theta = \frac{5\pi}{6} + 2\pi n \quad (n \in \mathbb{Z})$$
6. 3. Substitute back and solve for $x$ for both solution sets:
7. $$3x - \frac{\pi}{6} = \frac{\pi}{6} + 2\pi n \implies x = \frac{\pi}{9} + \frac{2\pi n}{3}$$
8. $$3x - \frac{\pi}{6} = \frac{5\pi}{6} + 2\pi n \implies x = \frac{\pi}{3} + \frac{2\pi n}{3}$$
9. 4. Filter for $0 \leq x < 2\pi$ to get 6 total valid solutions:
10. $\frac{\pi}{9}, \frac{7\pi}{9}, \frac{13\pi}{9}, \frac{\pi}{3}, \pi, \frac{5\pi}{3}$

> **Exam tip:** Always check the range of the isolated trigonometric term first—if it's outside the possible range (e.g., $\cos x = 2$), you can immediately conclude no real solutions exist, saving time on multiple-choice questions.

## Solving Quadratic Trigonometric Equations

**Quadratic Trigonometric Equation** — An equation that can be rewritten as a quadratic polynomial in a single trigonometric function $f(\theta)$, where $a \neq 0$.

*Notation:* $af^2(\theta) + bf(\theta) + c = 0$

To solve these equations, first use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to convert mixed terms into a single-variable quadratic. Next, solve for the trigonometric function by factoring or the quadratic formula. Each solution becomes a separate linear trigonometric equation, which you solve using the linear method.

**Worked example:** Find all solutions to $2\cos^2 x - 3\sin x = 0$ for $0 \leq x < 2\pi$.

1. 1. Use the Pythagorean identity $\cos^2 x = 1 - \sin^2 x$ to rewrite in terms of $\sin x$:
2. $$2(1 - \sin^2 x) - 3\sin x = 0 \implies 2\sin^2 x + 3\sin x - 2 = 0$$
3. 2. Factor the quadratic by substituting $u = \sin x$:
4. $$2u^2 + 3u - 2 = (2u - 1)(u + 2) = 0$$
5. This gives solutions $u = \frac{1}{2}$ and $u = -2$.
6. 3. $\sin x = -2$ is impossible (range of sine is $[-1,1]$), so discard it. For $\sin x = \frac{1}{2}$:
7. Solutions on $0 \leq x < 2\pi$ are $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$.
8. 4. Verify by substituting back into the original equation:
9. $$2\left(\frac{\sqrt{3}}{2}\right)^2 - 3\left(\frac{1}{2}\right) = \frac{3}{2} - \frac{3}{2} = 0 \quad \checkmark$$

> **Exam tip:** When you use an operation that can introduce extraneous solutions (like squaring both sides of an equation), always verify your solutions by plugging them back into the original equation.

## Solving Trigonometric Inequalities

A trigonometric inequality requires finding all input values that make a trigonometric expression satisfy an inequality relationship. We leverage continuity and periodicity of trigonometric functions to find valid solution intervals.

The core workflow is: 1. Solve the corresponding equality (replace the inequality with an equals sign) to find all critical points; 2. Sort critical points in increasing order; 3. Test the sign of the expression in each interval between critical points, or use unit circle symmetry to identify the valid region directly.

**Worked example:** Find all solutions to $\cos(2x) \geq \frac{1}{2}$ on the domain $0 \leq x < 2\pi$.

1. 1. Solve the corresponding equality $\cos(2x) = \frac{1}{2}$. Let $\theta = 2x$:
2. $$\cos \theta = \frac{1}{2} \implies \theta = \frac{\pi}{3} + 2\pi n, \quad \theta = \frac{5\pi}{3} + 2\pi n$$
3. Substitute back $\theta = 2x$ and solve for $x$:
4. $$x = \frac{\pi}{6} + \pi n, \quad x = \frac{5\pi}{6} + \pi n$$
5. 2. Filter and sort critical points for $0 \leq x < 2\pi$: $\frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$.
6. 3. Use unit circle symmetry: $\cos \theta \geq \frac{1}{2}$ when $-\frac{\pi}{3} + 2\pi n \leq \theta \leq \frac{\pi}{3} + 2\pi n$. Substitute back $\theta = 2x$:
7. $$-\frac{\pi}{6} + \pi n \leq x \leq \frac{\pi}{6} + \pi n$$
8. 4. Filter for $0 \leq x < 2\pi$ to get the final solution:
9. $0 \leq x \leq \frac{\pi}{6} \cup \frac{5\pi}{6} \leq x \leq \frac{7\pi}{6} \cup \frac{11\pi}{6} \leq x < 2\pi$

> **Exam tip:** Remember that cosine corresponds to the $x$-coordinate and sine to the $y$-coordinate on the unit circle, so you can immediately identify which quadrants satisfy the inequality without testing every interval.

## Common pitfalls

- **Wrong:** Forgetting to divide the periodicity term by the coefficient of $x$, e.g., writing the general solution of $\sin(3x) = 1/2$ as $x = \pi/6 + 2\pi n$ instead of $x = \pi/18 + 2\pi n/3$.
  - Why it fails: Students often only add the periodicity for the original trigonometric function and forget to adjust for the horizontal stretch.
  - Correct: After solving for $\theta = bx$, divide the entire solution (including the periodicity term) by $b$ to get the solution for $x$.
- **Wrong:** Only finding one solution per period for sine or cosine, e.g., only writing $x = \pi/6$ for $\sin x = 1/2$ on $0 \leq x < 2\pi$, missing $x = 5\pi/6$.
  - Why it fails: Students stop after finding the reference angle from the inverse trigonometric function, and forget sine and cosine take each value twice per period.
  - Correct: After finding the first solution from inverse trigonometry, always use unit circle symmetry to find the second solution in $[0, 2\pi)$.
- **Wrong:** Adding $2\pi n$ instead of $\pi n$ to the general solution of tangent equations.
  - Why it fails: Students memorize $2\pi$ periodicity from sine and cosine, and forget tangent has a shorter period of $\pi$.
  - Correct: Always note which trigonometric function you are working with, and add $\pi n$ for tangent, $2\pi n$ for sine and cosine.
- **Wrong:** Keeping out-of-range solutions when solving quadratic trigonometric equations, e.g., keeping $u = -2$ as a valid solution for $\sin x = -2$.
  - Why it fails: Students focus on factoring the quadratic and forget to check if each solution is within the range of the trigonometric function.
  - Correct: After solving for the trigonometric function, check if any solution for sine or cosine is between $-1$ and $1$ before proceeding to find $x$.
- **Wrong:** Mismatching endpoint inclusion for inequalities, e.g., including endpoints for a strict inequality $f(x) > 0$.
  - Why it fails: Students rush to write the solution and don't check the original inequality sign.
  - Correct: Always match the inequality sign: include endpoints if the inequality is non-strict ($\leq, \geq$), exclude endpoints if it is strict ($<, >$).
- **Wrong:** Not checking for extraneous solutions after squaring both sides of a trigonometric equation.
  - Why it fails: Squaring introduces solutions that make the two sides negatives of each other, which satisfy the squared equation but not the original.
  - Correct: After solving any equation where you squared both sides, plug every solution back into the original equation to discard invalid solutions.

## Cheatsheet

| Category | Formula / Rule | Notes |
| --- | --- | --- |
| Linear trigonometric equation | $a f(bx + c) = d$, $f \in \{\sin, \cos, \tan\}$ | Isolate $f(bx + c)$ first, check $\|d/a\| \leq 1$ for $\sin, \cos$ |
| Periodicity for general solutions | Add $2\pi n$ ($n \in \mathbb{Z}$) for $\sin, \cos$; add $\pi n$ for $\tan$ | Always divide entire solution (including periodicity) by $b$ |
| Quadratic trigonometric standard form | $a f^2(x) + b f(x) + c = 0$ | Use Pythagorean identities to convert mixed terms to single variable |
| Solutions for $\sin \theta = k$ | $\theta = \arcsin(k) + 2\pi n$, $\theta = \pi - \arcsin(k) + 2\pi n$ | Only valid for $\|k\| \leq 1$, two solutions per $2\pi$ period |
| Solutions for $\cos \theta = k$ | $\theta = \arccos(k) + 2\pi n$, $\theta = -\arccos(k) + 2\pi n$ | Only valid for $\|k\| \leq 1$, two solutions per $2\pi$ period |
| Solutions for $\tan \theta = k$ | $\theta = \arctan(k) + \pi n$ | One solution per $\pi$ period, valid for all real $k$ |
| Trigonometric inequality workflow | 1. Solve equality for critical points<br>2. Sort critical points<br>3. Test sign in intervals | Match endpoint inclusion to original inequality sign |
| Pythagorean conversion | $\sin^2 x = 1 - \cos^2 x$, $\cos^2 x = 1 - \sin^2 x$ | Use to convert mixed quadratic terms to single variable |

## What's next

Mastering trigonometric equations and inequalities is a critical foundation for all remaining topics in AP Precalculus and future calculus courses. This topic is frequently combined with trigonometric identities, function transformations, and real-world modeling to create multi-step exam questions, so practicing the step-by-step methods outlined here will help you tackle even the most complex problems on test day. You can extend your understanding by exploring polar coordinates and parametric trigonometric functions, which rely on the same core solving skills, and by working through additional AP-style practice questions to build speed and accuracy for the exam.

- [Unit 3: Trigonometric and Polar Functions Overview](https://www.owlsprep.com/study/ap-precalculus-u3-overview/)
- [Trigonometric Identities (Prerequisite Review)](https://www.owlsprep.com/study/ap-precalculus-u3-trigonometric-identities/)
- [Equivalent representations of trigonometric functions](https://www.owlsprep.com/study/ap-precalculus-u3-equivalent-representations-of-trigonometric-functions/)

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