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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

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/√
2
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. 

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