Solving trigonometric equations

IB Mathematics: Applications and Interpretation HL· 20 min read

1. Linear Trigonometric Equations Over a Specified Domain★★☆☆☆⏱ 5 min

A linear trigonometric equation has the form $a sin (b x + c) + d = 0$ (or with cosine/tangent) where there is only one type of trigonometric term. To solve, first isolate the trigonometric function, use inverse trigonometry to find the principal solution, then find all other solutions in the domain using symmetry and periodicity.

📘 Definition

Principal Solution

The smallest non-negative solution of a trigonometric equation found directly from evaluating the inverse trigonometric function.

📐 Worked Example

Solve $2 sin θ - 1 = 0$ for $0 \leq θ < 2 π$

1
Rearrange the equation to isolate $sin θ$ :
2
$2\sin\theta = 1 \implies \sin\theta = \frac{1}{2}$
3
Find the principal solution using $arcsin (\frac{1}{2})$ :
4
$\theta = \frac{\pi}{6}$
5
Sine is positive in the first and second quadrants, so find the second solution using symmetry:
6
$\pi - \frac{\pi}{6} = \frac{5\pi}{6}$
7
Check that both solutions fall within the domain $0 \leq θ < 2 π$ . The period of $sin θ$ is $2 π$ , so there are no additional solutions.

Exam tip:

Always confirm how many solutions you expect based on the domain size and function period before writing your final answer.

2. General Solutions of Trigonometric Equations★★★☆☆⏱ 6 min

When no domain is specified, we use the periodicity of trigonometric functions to write a general solution that describes all possible solutions. We do this by adding an integer multiple of the period to each base solution.

Sine and cosine have a period of $2 π$ , while tangent has a period of $π$ . This changes the form of the general solution.

📐 Worked Example

Find the general solution of $cos x = \frac{3}{2}$

1
Find the two base solutions over one period $[0, 2 π)$ :
2
$x = \frac{\pi}{6} \quad \text{and} \quad x = \frac{11\pi}{6}$
3
Add an integer multiple of the period $2 π$ to each base solution:
4
$x = \frac{\pi}{6} + 2k\pi, \quad x = \frac{11\pi}{6} + 2k\pi$
5
This can be condensed into a single form, where $k$ is any integer:
6
$x = 2\pi k \pm \frac{\pi}{6}, \quad k \in \mathbb{Z}$

3. Quadratic Trigonometric Equations★★★☆☆HL only⏱ 7 min

A quadratic trigonometric equation can be rewritten in the form $a sin^{2} x + b sin x + c = 0$ (or for cosine). We almost always use the Pythagorean identity $sin^{2} x + cos^{2} x = 1$ to rewrite equations with multiple trigonometric terms into a single quadratic form, then solve like any standard quadratic.

📐 Worked Example

Solve $2 cos^{2} x - 3 sin x = 0$ for $0 \leq x < 2 π$

1
Substitute $cos^{2} x = 1 - sin^{2} x$ to rewrite the equation in terms of $sin x$ only:
2
$2(1 - \sin^2 x) - 3\sin x = 0 \implies 2 - 2\sin^2 x - 3\sin x = 0$
3
Rearrange into standard quadratic form:
4
$2\sin^2 x + 3\sin x - 2 = 0$
5
Let $u = sin x$ and factor the quadratic:
6
$2u^2 + 3u - 2 = (2u - 1)(u + 2) = 0 \implies u = \frac{1}{2} \text{ or } u = -2$
7
$sin x = - 2$ has no solution, since the range of sine is $[- 1, 1]$ . Solve $sin x = \frac{1}{2}$ :
8
$x = \frac{\pi}{6}, \quad \frac{5\pi}{6}$

4. Equations with Scaled Trigonometric Arguments★★★★☆⏱ 6 min

For a trigonometric term like $sin (k x)$ , the period is adjusted to $\frac{2 π}{∣ k ∣}$ for sine/cosine, and $\frac{π}{∣ k ∣}$ for tangent. A common strategy is to first adjust the domain to match the argument, solve, then filter for valid values of $x$ .

📐 Worked Example

Solve $tan (2 x) = 1$ for $0 \leq x < 2 π$

1
Adjust the original domain to match the argument $2 x$ :
2
$0 \leq x < 2\pi \implies 0 \leq 2x < 4\pi$
3
Write the general solution for $tan α = 1$ , where $α = 2 x$ :
4
$\alpha = \frac{\pi}{4} + k\pi \implies 2x = \frac{\pi}{4} + k\pi$
5
Rearrange to solve for $x$ :
6
$x = \frac{\pi}{8} + \frac{k\pi}{2}, \quad k \in \mathbb{Z}$
7
Substitute integer values of $k$ to get all solutions in the original domain:
8
$x = \frac{\pi}{8}, \frac{5\pi}{8}, \frac{9\pi}{8}, \frac{13\pi}{8}$

5. Common Pitfalls

Wrong move:

Stopping at only the principal solution, forgetting other solutions in the domain

Why:

Most linear trigonometric equations have 2 solutions per period, so you will lose marks for missing solutions

Correct move:

After finding the principal solution, always use quadrant symmetry to find the second base solution, then add/subtract the period to get all solutions in the domain

Wrong move:

Forgetting to adjust the domain for scaled arguments like $sin (3 x)$

Why:

If you do not adjust the domain to match the argument, you will miss half or more of the required solutions

Correct move:

Always adjust the domain to match the argument of the trigonometric function first, then filter back to the original $x$ domain after solving

Wrong move:

Discarding the entire quadratic solution because one root is invalid

Why:

Many quadratics give one root outside the range $[- 1, 1]$ for sine/cosine, but the other root is often valid

Correct move:

Check all roots from the quadratic, discard any that are outside the range of the trigonometric function, and solve for all valid roots

Wrong move:

Writing general solutions without stating $k$ is an integer

Why:

Examiners require explicit confirmation that you understand the general form describes infinitely many solutions

Correct move:

Always add $k \in Z$ (or state $k$ is any integer) after writing a general solution

6. Quick Reference Cheatsheet

Equation Type	General Solution Form	Key Note
$a sin (b x + c) = d$	$b x + c = arcsin (d / a) + 2 k π$ ; $π - arcsin (d / a) + 2 k π$	Check $∣ d / a ∣ \leq 1$
$a cos (b x + c) = d$	$b x + c = \pm arccos (d / a) + 2 k π$	Check $∣ d / a ∣ \leq 1$
$a tan (b x + c) = d$	$b x + c = arctan (d / a) + k π$	Tangent period = $π$ , not $2 π$
Quadratic trig	Rewrite with Pythagorean identity, solve quadratic	Discard roots outside $[- 1, 1]$

7. Frequently Asked

Do I need to give general solutions if a domain is specified?

No. If a domain is explicitly given, you only need to list all solutions that fall within that domain. Only provide the general form when explicitly asked for it.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · Paper 1
Solve sin equation over 0 to 2π
2022 · Paper 2
Quadratic trig equation in context
2023 · Paper 1
Find general solution of cos equation

Going deeper

syllabusIB AI HL Official SyllabusOfficial subject reference

What's Next

Solving trigonometric equations is a foundational skill for almost all applied trigonometry topics in IB AI HL. You will use these methods to solve for unknown parameters in trigonometric models of periodic phenomena, like tides, seasonal temperatures, and wave motion. These skills also underpin calculus operations on trigonometric functions, which appear in both paper 1 and paper 2 exams. Mastery of this sub-topic is critical for scoring well on trigonometry-related questions, which appear frequently in both SL and HL assessments.