# Solving trigonometric equations

> IB Mathematics: Applications and Interpretation HL · Geometry and trigonometry
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u3-solving-trigonometric-equations/

This module covers solving linear trigonometric equations over a specified finite domain. You will learn to find the principal solution and use the symmetry and period of sine, cosine and tangent to list every solution that lies within a given interval.

**Prerequisites:** [Trigonometric identities and exact values](https://www.owlsprep.com/study/ib-math-ai-hl-u3-trigonometric-identities/); [Unit circle and periodicity of trig functions](https://www.owlsprep.com/study/ib-math-ai-hl-u3-trigonometric-functions/)

## Learning objectives

- Solve linear trigonometric equations over a specified finite domain
- Use the period and symmetry of a trigonometric function to find all solutions within a given interval
- Use a GDC to solve sinusoidal model equations graphically over a specified interval
- Recognise when a trigonometric equation has no solution

## Linear Trigonometric Equations Over a Specified Domain

A linear trigonometric equation has the form $a\sin(bx + c) + d = 0$ (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.

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

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

1. Rearrange the equation to isolate $\sin\theta$:
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 $0 \leq \theta < 2\pi$. The period of $\sin\theta$ is $2\pi$, 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.

## Solving Sinusoidal Model Equations with a GDC

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.

> **info**
>
> Set your GDC to radians when the model uses $\pi$ in the coefficient. Reading solutions off the graph, rather than deriving a general solution, is the standard AI approach and is fully acceptable in the mark scheme.

**Worked example:** A seat on a Ferris wheel has height above the ground, in metres, modelled by $h(t) = 8\sin\left(\frac{\pi}{5} t\right) + 10$, 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 $y = 8\sin\left(\frac{\pi}{5}x\right) + 10$ and the horizontal line $y = 14$ over $0 \leq x \leq 20$.
4. The period is $\frac{2\pi}{\pi/5} = 10$ 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.

## Common pitfalls

- **Wrong:** Stopping at only the principal solution, forgetting other solutions in the domain
  - Why it fails: Most linear trigonometric equations have 2 solutions per period, so you will lose marks for missing solutions
  - Correct: 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:** Solving a model equation on the GDC while it is set to degrees, when the coefficient uses radians
  - Why it fails: A model such as $\sin\left(\frac{\pi}{5}t\right)$ is written in radians, so degree mode gives completely wrong intersection values
  - Correct: 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

## 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.

- [Vectors: basics, operations, dot product](https://www.owlsprep.com/study/ib-math-ai-hl-u3-vectors-basics-operations-dot-product/)
- [Vector Equation of a Line, Intersections](https://www.owlsprep.com/study/ib-math-ai-hl-u3-vector-equation-of-a-line/)
- [Voronoi diagrams: basic construction](https://www.owlsprep.com/study/ib-math-ai-hl-u3-voronoi-diagrams-basic-construction/)

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