Solving trigonometric equations

Q: 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.

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. Solving Sinusoidal Model Equations with a GDC★★★☆☆⏱ 6 min

In IB AI HL, trigonometric equations most often arise from a periodic model such as a Ferris wheel, tide height, or temperature. You are given a function like $h (t) = a sin (b (t - c)) + d$ and asked for the time(s) at which the model reaches a particular value over a specified interval. Since AI uses a GDC throughout, the expected method is graphical: plot the model and the target value, then use the calculator to find every intersection in the interval.

📐 Worked Example

A seat on a Ferris wheel has height above the ground, in metres, modelled by $h (t) = 8 sin (\frac{π}{5} t) + 10$ , where $t$ is the time in seconds after the wheel starts. Find all times in the interval $0 \leq t \leq 20$ at which the seat is exactly 14 m above the ground.

1
Write the equation you need to solve by setting the model equal to the target height:
2
$8\sin\left(\frac{\pi}{5} t\right) + 10 = 14$
3
On the GDC (in radian mode), graph $y = 8 sin (\frac{π}{5} x) + 10$ and the horizontal line $y = 14$ over $0 \leq x \leq 20$ .
4
The period is $\frac{2 π}{π /5} = 10$ seconds, so the interval $0 \leq t \leq 20$ contains two full periods. Expect the horizontal line to cross the curve four times.
5
Use the calculator's intersect feature to read off each crossing:
6
$t \approx 0.83, \quad t \approx 4.17, \quad t \approx 10.83, \quad t \approx 14.17 \text{ seconds}$
7
All four values lie in the interval, so these are the required times (to 3 significant figures).

Exam tip:

Sketch the target line first and count the intersections against the number of periods in the interval. That tells you how many solutions to look for, so you do not stop early after finding just one or two.

3. 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:

Solving a model equation on the GDC while it is set to degrees, when the coefficient uses radians

Why:

A model such as $sin (\frac{π}{5} t)$ is written in radians, so degree mode gives completely wrong intersection values

Correct move:

Set the GDC to radian mode before graphing, and count the intersections against the number of periods in the interval to be sure you have found them all

4. 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.

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.