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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

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

Police car


Are you a budding crime scene officer?  Do you love watching CSI and programmes like that?

Did you know that at times trigonometry (trig) is used in crime scenes?

Maybe the investigators want to know the trajectory of something, or the distance, or the angle.  


story bookTime for a little story


Once upon a time students studying A level needed to know the exact values of trig.

They may have wanted careers that involved maths at a higher level, or they may just have liked solving puzzles.

Along came someone who said, A level students shouldn't have all the fun, let G.C.S.E students join in.

Is it a crime?  Up to you to decide!


Let's investigate

The exact values of Sin, Cos and Tan that you need to know are those for 0°, 30°, 45° 60° and 90° for each ratio.

There are a few ways of calculating them.


You may have been taught this way.


equilateral triangleEquilateral triangle split into two right angles. right angled triangle


Taking an equilateral triangle, giving each side a unit of 2, and splitting it into a right-angled triangle as shown.

You would then label the triangle's hypotenuse, opposite and adjacent in order to find the trig values of Sin, Cos or Tan.

e.g Sin 30° = 1/2  Sin 60° = √3/2


Isosceles triangle


or an isosceles triangle to do the same.  

If you are secure in labelling the triangles and using the trig ratio then this is absolutely fine. 

It can get quite messy having to keep labelling the triangles (opposites and adjacent are not always in the same place)


A different way without having to split and label triangles.

You know the trig values required  0°, 30°, 45° 60° and 90° for each ratio. 

You need the numbers 0,1,2,3,4 and be able to simplify square roots if you can.


Exact values of sin

To find exact values of sin √angle ÷ 2

Look at the numbers at the top of the table ranging from 0 to 4 this is assigned to your √

exact values sin

Looking at the example above all the numbers assigned to the angles have been √ ÷ 2 and where possible simplified. 

√ can only be simplified to a whole number, for example, √4 = 2


Exact Values of Cos

exact values of Cos

To find exact values of Cos - √angle ÷ 2

The numbers at the top of the table go from 4 to 0.

What do you notice about the answers?


Exact values of tan

exact values of trig ratios

This is slightly different.  Values of sin ÷ values of cos and simplify if possible.

Note that tan 90º can't be done.  If you put 1 ÷ 0 into your calculator it comes up as an error.


During the activities, try to recreate the table for yourself, rather than refer to them as it will help with understanding the calculations.


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