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Find the Angle Using Trigonometry

In this worksheet, students will use trigonometry to calculate the angle in right-angled triangles.

'Find the Angle Using Trigonometry' worksheet

Key stage:  KS 3

Year:  Year 9 Maths worksheets

Curriculum topic:   Geometry and Measures

Curriculum subtopic:   Solve Right-Angled Triangle Problems

Popular topics:   Angles worksheets, Geometry worksheets

Difficulty level:  

Worksheet Overview

In a right-angled triangle, the sides and the angles are connected by three trigonometric ratios.



The three trig ratios are:



The best way to use these is in a formula triangle:


For example with sin:


sine formula triangle


The triangle above shows the formula triangle for using sin. 

To use it, we simply cover up the part of the triangle that we want to find out and then rearrange the equation.


For example, if we know the sin and the opposite side, we cover up H (the hypotenuse) and we can see that we need to do: O ÷ S = H see below


sine formula triangle


If we know the sin and the hypotenuse, we cover up O and we can do this equation: S x H = O see below:


sine formula triangle


Clearly, if we know O and A but don't know S, we'd cover up S and do O ÷ H = S


We can remember the triangle by using the letters SOH to spell a sort of word!


We can do the same for the other two formulae too:


Here is the triangle for cos - we can remember this one by the word CAH


cosine triangle


And here's the one for tan - we can remember this one by the word TOA


tangent triangle


We can stick the words together to make a pretty silly-looking word:





Example to find an angle

Using trigonometry, calculate the angle θº to 3 sig. figs.




Label the sides adjacent, opposite and hypotenuse in relation to the angle θº.

The given sides are the adjacent and the hypotenuse.


Which formula do we need that includes the angle, adjacent and hypotenuse? It is CAH




We know the adjacent and the hypotenuse, so in the formula triangle, we need to cover up C.


This gives us:

 cos angle = adjacent ÷ hypotenuse


cos θ = 3 ÷ 5 = 0.6


To find the angle itself, we need to do the inverse of cos.

First, make sure that your calculator is in degrees mode.


θ = cos-1 (0.6)

θ = 53.1º (3 s.f.)


Let's have a go at some questions.

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