---
title: "Solving Trigonometric Equations in a Given Interval"
description: "Find every solution inside a stated interval, using the graph or the circle to get the solutions the calculator does not show, and adjusting the interval first when the angle is 2x or x + 30°."
canonical: https://lightmysky.com/learn/mathematics/solving-trigonometric-equations-in-a-given-interval-mt_74_jiK2xXA
source: https://lightmysky.com/learn/mathematics/solving-trigonometric-equations-in-a-given-interval-mt_74_jiK2xXA.md
retrieved: 2026-09-12
---

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# Solving Trigonometric Equations in a Given Interval

Find every solution inside a stated interval, using the graph or the circle to get the solutions the calculator does not show, and adjusting the interval first when the angle is 2x or x + 30°.

Subject: Mathematics · Area: Geometry · Ages 16 to 17
Page: https://lightmysky.com/learn/mathematics/solving-trigonometric-equations-in-a-given-interval-mt_74_jiK2xXA

## Ready when they can

- Solve sin x = 0.5 for 0 ≤ x ≤ 2π and give both solutions
- Stretch the interval before solving an equation written in 2x
- Use an identity to reduce an equation to a single trigonometric function first

## Lesson: Finding every solution in the interval

Most heights are hit twice per turn. Sin x equals 0.5 gives pi over 6 and 5 pi over 6, mirror images across pi over 2. Find the reference angle first, then place both. One answer alone is half marks.

**Example.** For sin 2x equals 0.5 with x between 0 and 2 pi, stretch first. Then 2x runs from 0 to 4 pi, covering two full turns. Solve for 2x, then halve each hit: pi over 12, 5 pi over 12, 13 pi over 12, 17 pi over 12.

Mixed equations must become one function first. Replace cos squared with 1 minus sin squared. The result factors like a quadratic in sin x. Each factor gives its own angles, so gather every one in range.

**Tip.** Never divide by a trig function, since that deletes solutions. Check each answer by substitution. Sketch the wave or the circle to confirm both twins are placed.

**Recap.** Place both twins, stretch the interval first, reduce to one function, and check each answer.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [The Pythagorean and Quotient Identities](https://lightmysky.com/learn/mathematics/the-pythagorean-and-quotient-identities-mt_NXBq37vxQh)
- [Solving Linear Inequalities and Solution Sets](https://lightmysky.com/learn/mathematics/solving-linear-inequalities-and-solution-sets-mt_QkG3Cw5VgM)
- [The Unit Circle and the Graphs of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/the-unit-circle-and-the-graphs-of-sine-cosine-and-tangent-mt_VJaimevi3-)

## Opens up

- [Motion on a Rough Inclined Plane](https://lightmysky.com/learn/mathematics/motion-on-a-rough-inclined-plane-mt_0rBy66HFNO)
- [Compound Angle Formulae](https://lightmysky.com/learn/mathematics/compound-angle-formulae-mt_ANDmX8exwg)