Exact Trig Values (AQA GCSE Maths)

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:

• 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
  • The angles will be 90°, 60° and 30°
  • Using SOHCAHTOA gives…
    • Sin 60° =               Sin 30°  =
    • Cos 60°  =                  Cos 30° =
    • Tan 60°  =                Tan 30° = =
• 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
  • The two equal angles will be 45°
  • Using SOHCAHTOA gives…
    • Sin
    • Cos
    • Tan = 1