Finding missing sides with trigonometry
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
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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 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).