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 should be familiar with these values and be able to derive the values using geometry
You are expected to know the exact values of sin, cos and tan for angles of 0°, 30° , 45°, 60°, 90°, 180° and their multiples
The exact values you are expected to know are summarised here:

Trigonometry Exact Values RN table, downloadable IGCSE & GCSE Maths revision notes

Note that the values of sin θ going from 0° to 90° match those of cos θ going from 90° to 0°

How are exact values in trigonometry derived?

There are two special right-angled triangles that can be used to derive all of the exact values you need to know
Consider a right-angles triangle with a hypotenuse of 2 units and a shorter side length of 1 unit
- Using Pythagoras’ theorem the third side will be $begin mathsize 16px style square root of 3 end style$
- The angles will be 90°, 60° and 30°
- Using SOHCAHTOA gives…
  - Sin 60° = $begin mathsize 16px style fraction numerator square root of 3 over denominator 2 end fraction end style$ Sin 30° = $1 half$
  - Cos 60° = $1 half$ Cos 30° = $fraction numerator square root of 3 over denominator 2 end fraction$
  - Tan 60° = $square root of 3$ Tan 30° = $fraction numerator 1 over denominator square root of 3 end fraction$ = $begin mathsize 16px style fraction numerator square root of 3 over denominator 3 end fraction end style$
Consider an isosceles triangle with two equal side lengths (the opposite and adjacent) of 1 unit
- Using Pythagoras’ theorem it will have a hypotenuse of $square root of 2$
- The two equal angles will be 45°
- Using SOHCAHTOA gives…
  - Sin $begin mathsize 16px style 45 degree space equals space fraction numerator 1 over denominator square root of 2 end fraction space equals space fraction numerator square root of 2 over denominator 2 end fraction end style$
  - Cos $begin mathsize 16px style 45 degree space equals space fraction numerator 1 over denominator square root of 2 end fraction space equals space fraction numerator square root of 2 over denominator 2 end fraction end style$
  - Tan $45 degree$ = 1

Exam Tip

You will be expected to be comfortable using exact trig values for certain angles but it can be easy to muddle them up if you just try to remember them from a list
sketch the triangles and trig graphs on your paper so that you can use them as many times as you need to during the exam
- sketch the triangles for the key angles $45 degree$ $30 degree$ , $60 degree$
- sketch the trig graphs for the key angles $0 degree$ , $90 degree$ , $180 degree$ , $270 degree$ , $360 degree$

Worked example

Using an equilateral triangle of side length 2 units, derive the exact values for the sine, cosine and tangent of 60° and 30°.

Sketch the triangle and create two right angled triangles by drawing the line of symmetry through the middle.

Use Pythagoras’ theorem to find the vertical height of the triangle.

$B D space equals space square root of 2 to the power of 2 space end exponent minus space 1 squared end root space equals space square root of 3$

Use SOHCAHTOA to find the trig ratios for 30° and 60°.

Sin 60° = $fraction numerator bold italic B bold italic D over denominator bold italic A bold italic B end fraction bold equals fraction numerator square root of bold 3 over denominator bold 2 end fraction$ Sin 30° = $fraction numerator bold italic A bold italic D over denominator bold italic A bold italic B end fraction bold equals bold space bold 1 over bold 2$

Cos 60° = $fraction numerator bold italic A bold italic D over denominator bold italic A bold italic B end fraction bold equals bold space bold 1 over bold 2$ Cos 30° = $fraction numerator bold italic B bold italic D over denominator bold italic A bold italic B end fraction bold equals bold space fraction numerator square root of bold 3 over denominator bold 2 end fraction$

Tan 60° = $fraction numerator bold B bold D bold space over denominator bold A bold D end fraction bold equals fraction numerator bold space square root of bold 3 over denominator bold 1 end fraction bold space bold equals bold space square root of bold 3$ Tan 30° = $begin mathsize 16px style fraction numerator bold italic A bold italic D over denominator bold italic B bold italic D end fraction bold equals fraction numerator bold 1 over denominator square root of bold 3 end fraction end style$ = $begin mathsize 16px style fraction numerator square root of bold 3 over denominator bold 3 end fraction end style$

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