Exact Trig Values | Mathematics for GCSE/IGCSE - Year 11 PDF Download

What are exact values in trigonometry?

• Exact values in trigonometry refer to specific values of sine, cosine, and tangent for certain angles that can be expressed precisely using fractions and surds.
  • You should be able to recognize and derive these values geometrically.
• It is essential to know the exact values 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 provide exact values for trigonometric functions.

• Consider a right-angled triangle with a hypotenuse of 2 units and a shorter side length of 1 unit. Applying Pythagoras’ theorem, the third side will be √3 units. The angles are 90°, 60°, and 30°. Utilizing SOHCAHTOA:
  • Sin 60° = √3/2, Sin 30° = 1/2
  • Cos 60° = 1/2, Cos 30° = √3/2
  • Tan 60° = √3, Tan 30° = (√3)/3
• Now, consider an isosceles triangle with two equal side lengths (the opposite and adjacent) of 1 unit. Applying Pythagoras’ theorem, its hypotenuse will be √2 units. The two equal angles are 45°. Applying SOHCAHTOA:
  • Sin 45° = 1/√2 = √2/2
  • Cos 45° = 1/√2 = √2/2
  • Tan 45° = 1