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Find an Angle Using the Cosine Rule

In this worksheet, students will apply the cosine rule to find out the size of angles within a non-right-angled triangle.

'Find an Angle Using the Cosine Rule' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR,

Curriculum topic:   Geometry and Measures, Mensuration

Curriculum subtopic:   Mensuration and Calculation Triangle Mensuration

Difficulty level:  

Worksheet Overview

Ever wondered what it's like to dangle outside of the space station to go to work.




The robotic arms are operated by controlling the angles of its joints.  Calculating the final position of the astronaut involves trigonometry, one aspect of which is finding angles.


Things have come a long way since man first landed on the moon, who would have thought that studying angles could get us this job.

Did your careers advisor mention this as a future job?  I suspect not.

In preparation for our chosen career, we had better get to work.


The cosine rule is used to find angles in a non-right angled triangle.  And here it is.


Three for the price of one, not always the best deal!


Try this, and you won't have to worry about what a, b or c actually is.


cosine rule simplified


This looks similar to that of finding a side using the cosine rule.

Make sure your calculator is set to degrees.


Example 1

cosine rule find angle example


We are given two side lengths and the trapped angle.


Square one side, (6²) and add the square(5²) of the other side.

Subtract the square of the side that is opposite the angle. (7²)

Then multiply 2 by the side lengths next to the angle. e.g 2 x 6 x 5

Divide your answers.

As you want to find an angle you need Cos-1 



Example 2

cosine rule find an angle example


NOTE.  You can see here that you have a negative Cos  - 0.276. This means that the angle you convert to will be obtuse i.e over 90°  This is a good way to check if your answer is sensible.


Can you see that all you need are the two sides with the trapped angle? 

When want to convert to find the angle, you will always use the COS-1 button.

As long as you remember the general formula you need not worry about a,b, or c                      


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