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# Finding missing sides with trigonometry
- URL: https://www.esheets.io/finding-missing-sides-with-trigonometry/
- Published: 2025-01-05T19:25:06.000Z
- Updated: 2026-06-21T16:42:08.000Z
- Author: Richard Linnington
- Tags: Maths, Geometry and Shape

Trigonometry helps us solve real-world problems, like finding the height of a building or the distance across a river, without measuring them directly. By using the angles and a known side of a right triangle, we can calculate the missing sides with tools like sine, cosine, and tangent. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise finding missing sides with trigonometry with this self-marking maths worksheet.

The interactive worksheet below generates questions, gives instant feedback, and lets students record their score.

#### What you’ll practise

- Identifying the opposite, adjacent and hypotenuse sides.
- Choosing sine, cosine or tangent.
- Substituting known values into the correct ratio.
- Solving for the missing side.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

Answers should be correctly rounded to 1 decimal place.

  
## Topic guide

### What this worksheet practises

This worksheet provides practice on finding the lengths of missing sides in right-angled triangles using trigonometry (SOH CAH TOA). This is used when you know one angle and one side, and need to find a second side.

### Key method

Use the SOH CAH TOA acronym to identify the correct equation, and then use algebra to solve it.

- Label the sides: Hypotenuse (longest), Opposite (across from the known angle), and Adjacent (next to the known angle).
- Identify the side you **know**, and the side you **want to find**.
- Choose the ratio (Sin, Cos, or Tan) that contains those two specific sides.
- Set up the equation (e.g. cos(angle) = A/H).
- Use algebra to isolate the unknown side. (If the unknown is on top, multiply. If the unknown is on the bottom, swap it with the trig function).

### Worked example

**A right-angled triangle has an angle of 40°. The hypotenuse is 12cm. Find the length of the adjacent side.**

Step 1: We know the Hypotenuse (H) and want the Adjacent (A).

Step 2: SOH CAH TOA tells us that A and H means we must use Cos.

Step 3: Set up the equation.

cos(40°) = A / 12

Step 4: Solve for A. The unknown is on top, so we multiply.

A = 12 × cos(40°)

Step 5: Calculate the result.

A = 9.19 cm (to 2 d.p.)

### Common mistakes to avoid

The most frequent error occurs when the unknown side is on the bottom of the fraction (e.g. sin(30) = 5/H). Students often try to multiply (5 × sin30), which is wrong. If the unknown is on the bottom, you must divide the number by the trig function: H = 5 ÷ sin(30).

### How to check your answer

Always remember that the hypotenuse is the longest side of a right-angled triangle. If you calculate an Opposite or Adjacent side and it comes out larger than the Hypotenuse, you have definitely made a mistake (usually multiplying when you should have divided).