Exact trig values

For 30°, 45° and 60° the trig ratios have exact values using surds. They come from two triangles: half a square and half an equilateral triangle.

Key facts

  • sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3.
  • sin 45° = cos 45° = √2/2, tan 45° = 1.
  • sin 60° = √3/2, cos 60° = ½, tan 60° = √3.
  • sin 0° = 0, cos 0° = 1, sin 90° = 1, cos 90° = 0.

Worked examples

Example 1

Find the exact value of sin 60°.

  1. Take an equilateral triangle with sides 2 and cut it in half.
  2. The halves have sides 1, 2 and √3 (Pythagoras).
  3. sin 60° = opposite/hypotenuse = √3/2

Answer: √3/2

Example 2

Work out cos 30° × tan 60° exactly.

  1. √3/2 × √3
  2. = 3/2

Answer: 3/2