Loading please wait

The smart way to improve grades

Comprehensive & curriculum aligned

Try an activity or get started for free

Apply Exact Trig Values

In this worksheet, students will learn how to calculate the exact values of trigonometry.

'Apply Exact Trig Values ' worksheet

Key stage:  KS 4

Year:  GCSE

GCSE Subjects:   Maths

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

Curriculum topic:   Geometry and Measures, Mensuration

Curriculum subtopic:   Mensuration and Calculation Triangle Mensuration

Difficulty level:  

Worksheet Overview

There are some questions in Higher GCSE exam papers where you need to calculate lengths of triangles using trigonometry without a calculator.

The only way to solve these types of question is to learn exact trig values.


The values of sin, cos and tan that you need to know are those for 0°, 30°, 45° 60° and 90°.


Deriving the values

Take an equilateral triangle, giving each side a unit of 2:


equilateral, 2 by 2 by 2


Now cut it in half to make a right-angled triangle as shown:


lengths are now 1, 2, and root 3


Using Pythagoras, we can deduce that the vertical side is 3.


We now have a right-angled triangle with three known lengths and angles. It means we can apply sin, cos, and tan to work out their values for 30° and 60° by considering opposite, adjacent, and hypotenuse sides and using SOH CAH TOA


sin 60 = √3/2
cos 60 = 1/2
tan 60 = √3/1

sin 30 = 1/2
cos 30 = √3/2
tan 30 = 1/√3


Now, consider a right-angled isosceles triangle as shown:


Lengths of 1, 1, and root 2


We can give the two shorter lengths a unit of 1, and using Pythagoras, deduce the hypotenuse is √2.


Once again, we can use SOH CAH TOA, this time to work out all the values for 45°:

sin 45 = 1/√2
cos 45 = 1/√
tan 45 = 1/1



Learning the table

The table below summarises all the values you need to learn.

Each value has been simplified and rationalised.


  sin cos tan
0 1 0
30° 1/2 √3/2 √3/3
45° √2/2 √2/2 1
60° √3/2 1/2 √3
90° 1 0 -


As well as the values we calculated from our triangles above, you also need to be familiar with sin, cos, and tan of 0° and 90°.

Tips for remembering the table

The columns for sin and cos are the same list of values, but in reverse order.


 cos is the only trig ratio that doesn't have a value of zero at 0°.  


tan x = sin x / cos x. This means you can derive any value for tan by dividing sin and cos of the same angle. (e.g. tan 60 = sin 60 / cos 60 = √3/2 ÷ 1/2 = √3)


 tan 90 has no value because it would require us to divide 1 by 0, which is infinity. 


This is a tricky activity but it will become easier if you practise doing some questions.


girl thinking

What is EdPlace?

We're your National Curriculum aligned online education content provider helping each child succeed in English, maths and science from year 1 to GCSE. With an EdPlace account you’ll be able to track and measure progress, helping each child achieve their best. We build confidence and attainment by personalising each child’s learning at a level that suits them.

Get started

Try an activity or get started for free

  • National Tutoring Awards 2023 Shortlisted / Parents
    National Tutoring Awards 2023 Shortlisted
  • Private-Tutoring-WINNER-EducationInvestor-Awards / Parents
    Winner - Private Tutoring
  • Bett Awards Finalist / Parents
  • Winner - Best for Home Learning / Parents
    Winner - Best for Home Learning / Parents