Exact Trig Values (AQA GCSE Maths) : Revision Note

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Exact Trig Values

What are exact values in trigonometry?

  • For certain angles the values of sin θ, cos θ and tan θ can be written exactly

    • This means using fractions and surds

  • You are expected to know the exact values of sin, cos and tan for

    • 0°, 30°, 45°, 60°, 90°, 180° and their multiples

theta

30º

45º

60º

90º

sin theta

0

1 half

fraction numerator square root of 2 over denominator 2 end fraction

fraction numerator square root of 3 over denominator 2 end fraction

1

cos theta

1

fraction numerator square root of 3 over denominator 2 end fraction

fraction numerator square root of 2 over denominator 2 end fraction

1 half

0

tan theta

0

fraction numerator square root of 3 over denominator 3 end fraction

1

square root of 3

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How can I remember these exact trig values?

  • Look at patterns in the table

    • Note the values of sin θ from 0° to 90° match cos θ in reverse, from 90° to 0°

    • Some people remember sin θ using the trick fraction numerator square root of 0 over denominator 2 end fraction comma fraction numerator square root of 1 over denominator 2 end fraction comma fraction numerator square root of 2 over denominator 2 end fraction comma fraction numerator square root of 3 over denominator 2 end fraction comma fraction numerator square root of 4 over denominator 2 end fraction which simplifies to 0 comma 1 half comma fraction numerator square root of 2 over denominator 2 end fraction comma fraction numerator square root of 3 over denominator 2 end fraction comma 1

  • Two special right-angled triangles below can help you to find the exact values for 30º, 45º and 60º

    • Remember that by rationalising the denominator, fraction numerator 1 over denominator square root of 2 end fraction equals fraction numerator 1 over denominator square root of 2 end fraction cross times fraction numerator square root of 2 over denominator square root of 2 end fraction equals fraction numerator square root of 2 over denominator 2 end fraction

Exact Values Notes Diagram 1, A Level & AS Maths: Pure revision notes
Exact Values Notes Diagram 2, A Level & AS Maths: Pure revision notes
  • The trig graphs can help you to remember the exact values for 0º and multiples of 90º

y-equals-sinx
y-equals-cosx
y-equals-tanx

How do I use exact trig values?

  • You may come across trig questions in a non-calculator question

  • In trig calculations, substitute in the exact trig values and solve as usual

    • E.g. Solve the equation cos space 45 equals x over 12

      • Replace cos space 45 with fraction numerator square root of 2 over denominator 2 end fraction to give fraction numerator square root of 2 over denominator 2 end fraction equals x over 12

      • Then you can solve for x

  • On trig graphs, you may be expected to find a coordinate

    • E.g. The coordinates open parentheses 30 comma space k close parentheses lie on the graph y equals tan space x, find k

      • k will be equal to tan space 30

      • The exact value of tan space 30 is fraction numerator square root of 3 over denominator 3 end fraction

      • Therefore k equals fraction numerator square root of 3 over denominator 3 end fraction

Examiner Tips and Tricks

  • Writing these out (or sketching the triangles/graphs) on your paper at the beginning of the exam means that you can use them as many times as you need to during the exam!

Worked Example

Find the value of x in the diagram below.

Give your answer as an exact value.

Triangle ABC with angle BAC = 60º, AC = 28 cm and BC = x cm.

Triangle ABC is a right-angled triangle, so use SOHCAHTOA

We know the hypotenuse (AC) and we want to calculate the opposite (BC), so use sin space theta equals straight O over straight H

sin space 60 equals x over 28

Remember that sin space 60 equals fraction numerator square root of 3 over denominator 2 end fraction

So,

table row cell fraction numerator square root of 3 over denominator 2 end fraction end cell equals cell x over 28 end cell row cell 28 cross times fraction numerator square root of 3 over denominator 2 end fraction end cell equals x row cell 14 cross times square root of 3 end cell equals x end table

Leave in exact (surd) form

bold italic x bold equals bold 14 square root of bold 3 cm

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Amber

Author: Amber

Expertise: Maths Content Creator

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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