# Solving trigonometric equations

> IB Mathematics: Analysis and Approaches SL · Geometry & Trigonometry
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u3-solving-trigonometric-equations/

This sub-topic covers solving linear and quadratic trigonometric equations over restricted domains and for all real general solutions. You will learn to use periodicity and identities to find all valid solutions correctly for IB SL exams.

**Prerequisites:** [Trigonometric Pythagorean identities](https://www.owlsprep.com/study/ib-math-aa-sl-u3-trigonometric-identities/); [Unit circle and periodicity of trigonometric functions](https://www.owlsprep.com/study/ib-math-aa-sl-u3-trigonometric-functions/)

## Learning objectives

- Classify linear and quadratic trigonometric equations
- Solve trigonometric equations over a given restricted domain
- Find all general solutions of trigonometric equations over the reals
- Identify and avoid common errors when using periodicity

## Linear Trigonometric Equations over Restricted Domains

**Linear trigonometric equation** — An equation containing only one trigonometric term, raised to the first power, that can be rearranged to isolate the trigonometric term on one side.

*Notation:* $a\sin(bx + c) + d = k$

*Example:* $2\sin 3x = 1$

For IB AA SL, you will most often be given a restricted domain, a closed interval for $x$, and asked to find all solutions that lie within this interval. The core process is to isolate the trigonometric term first, then find all matching angles using the symmetry and periodicity of the function.

**Worked example:** Solve $2\sin x = 1$ for $0 \leq x < 2\pi$

1. Isolate the trigonometric term by dividing both sides by 2
2. $$\sin x = \frac{1}{2}$$
3. Find the principal solution using the unit circle or inverse sine
4. $$x = \arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}$$
5. Sine is positive in the first and second quadrants, so find the second solution in $[0, 2\pi)$
6. $$x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$$
7. Both solutions fall within the given domain, so these are the final results

> **Exam tip:** Always check that every solution you list falls inside the given domain; extra solutions outside the domain will cost you marks.

## General Solutions of Trigonometric Equations

When asked to find all solutions over $x \in \mathbb{R}$, you need to account for the repeating nature of trigonometric functions by adding a multiple of the function's period to your base solutions.

**General solution** — The complete set of all possible solutions, written by adding an integer multiple of the period to each base solution over one full period of the function.

> **info**
>
> The period of $\sin(bx)$ and $\cos(bx)$ is $\frac{2\pi}{|b|}$. The period of $\tan(bx)$ is $\frac{\pi}{|b|}$, regardless of $b$.

**Worked example:** Find the general solution of $2\sin x = 1$

1. Find the base solutions over one full period of $\sin x$ (period = $2\pi$)
2. $$x_1 = \frac{\pi}{6}, \quad x_2 = \frac{5\pi}{6}$$
3. Add an integer multiple of the period $2\pi$ to each base solution, where $k \in \mathbb{Z}$
4. $$x = \frac{\pi}{6} + 2k\pi, \quad x = \frac{5\pi}{6} + 2k\pi, \quad k \in \mathbb{Z}$$

## Quadratic Trigonometric Equations

Quadratic trigonometric equations have a squared trigonometric term, and can be solved with the same method as standard quadratics: substitute to get a quadratic in one variable, factor or use the quadratic formula, then solve each resulting linear trig equation separately.

**Worked example:** Solve $2\cos^2 x - 3\cos x + 1 = 0$ for $0 \leq x < 2\pi$

1. Substitute $u = \cos x$ to rewrite the equation as a standard quadratic
2. $$2u^2 - 3u + 1 = 0$$
3. Factor the quadratic and solve for $u$
4. $$(2u - 1)(u - 1) = 0 \implies u = \frac{1}{2} \text{ or } u = 1$$
5. Solve $\cos x = 1$ over the given domain
6. $$x = 0$$
7. Solve $\cos x = \frac{1}{2}$: cosine is positive in Q1 and Q4
8. $$x = \frac{\pi}{3}, \quad x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}$$
9. Collect all solutions that fall within the domain: $0, \frac{\pi}{3}, \frac{5\pi}{3}$

> **warning**
>
> Always discard any roots where $|\sin x| > 1$ or $|\cos x| > 1$ — these values are outside the range of the functions and have no solutions.

## Equations Requiring Identity Rearrangement

Many exam questions give equations with mixed trigonometric terms that require you to use identities (most often the Pythagorean identity) to rearrange into a solvable linear or quadratic form.

**Worked example:** Solve $2\sin^2 x = \cos x + 1$ for $0 \leq x < 2\pi$

1. Use the Pythagorean identity $\sin^2 x = 1 - \cos^2 x$ to rewrite the equation in terms of $\cos x$ only
2. $$2(1 - \cos^2 x) = \cos x + 1$$
3. Expand and rearrange into standard quadratic form
4. $$2\cos^2 x + \cos x - 1 = 0$$
5. Factor the quadratic
6. $$(2\cos x - 1)(\cos x + 1) = 0 \implies \cos x = \frac{1}{2} \text{ or } \cos x = -1$$
7. Solve each linear trig equation over the domain
8. $$\cos x = \frac{1}{2} \implies x = \frac{\pi}{3}, \frac{5\pi}{3}; \quad \cos x = -1 \implies x = \pi$$
9. All solutions are within the domain, so final solutions are $\frac{\pi}{3}, \pi, \frac{5\pi}{3}$

> **tip**
>
> Always substitute your solutions back into the original equation to check for extraneous solutions introduced by squaring or rearranging.

*Calculator:* forbidden

## Common pitfalls

- **Wrong:** Forgetting to add the period to every base solution when finding the general solution
  - Why it fails: Only adding the period to one solution leaves out half of the valid solutions for sine and cosine equations
  - Correct: Add the integer multiple of the period to every distinct base solution found over one full period
- **Wrong:** Canceling a common trigonometric term from both sides of an equation instead of factoring
  - Why it fails: Canceling removes all solutions where the trigonometric term equals zero, leading to missing solutions
  - Correct: Move all terms to one side of the equation and factor out the common trigonometric term
- **Wrong:** Including solutions that fall outside the given restricted domain
  - Why it fails: Adding or subtracting periods to base solutions creates solutions outside the required interval
  - Correct: List all candidate solutions, then filter out any that do not satisfy the domain inequality
- **Wrong:** Using the wrong period for scaled tangent functions like $\tan 2x$
  - Why it fails: Tangent has a period of $\pi$, not $2\pi$, so scaled tangent has half the period you might expect
  - Correct: For $\tan(bx)$, always use period $\frac{\pi}{|b|}$
- **Wrong:** Keeping solutions where $|\sin x| > 1$ or $|\cos x| > 1$
  - Why it fails: These values are outside the range of sine and cosine, so they have no corresponding real solutions
  - Correct: Discard any such roots immediately after solving for the trigonometric term

## Cheatsheet

| Equation Type | Solution Process |
| --- | --- |
| Linear | 1. Isolate the trigonometric term 2. Find all base solutions over one period 3. Add $k \times \text{period}$ for general solutions, filter for restricted domain |
| Quadratic | 1. Substitute $u = \sin x/\cos x/\tan x$ 2. Solve quadratic for $u$ 3. Discard values outside function range 4. Solve each resulting linear equation |
| Mixed (needs identity) | 1. Use Pythagorean/other identity to get one trig variable 2. Rearrange to linear/quadratic 3. Solve as above 4. Check for extraneous solutions |
| Period Rule | $\sin(bx), \cos(bx): T = \frac{2\pi}{\|b\|}$; $\tan(bx): T = \frac{\pi}{\|b\|}$ |

## What's next

Solving trigonometric equations is a core foundational skill that appears across all units of IB AA SL. You will use it to find critical points in calculus and solve for unknowns in periodic modeling problems. It is heavily tested in both Paper 1 and Paper 2 of your final exam, so practicing finding all solutions is critical for good marks. Next, you will apply this skill to triangle problems with the sine and cosine rule, then build on it to model real-world periodic phenomena with trigonometric functions.

- [Statistics & Probability](https://www.owlsprep.com/study/ib-math-aa-sl-u4-overview/)
- [Data types, sampling and bias](https://www.owlsprep.com/study/ib-math-aa-sl-u4-data-types-sampling-and-bias/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-aa-sl-u3-solving-trigonometric-equations/