Exact trigonometric values

An equilateral triangle of side 2 splits into two right-angled triangles with sides 1, √3 and 2. Their acute angles are 30° and 60°. A right-angled isosceles triangle with short sides 1 has hypotenuse √2 and acute angles 45°. Use opposite/hypotenuse for sine and adjacent/hypotenuse for cosine.

The values sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2 and cos 60° = 1/2 follow immediately. Both sin 45° and cos 45° equal √2/2. Tangent divides sine by cosine. Keep roots exact in a non-calculator solution; tangent at 90° is undefined because its denominator is zero.

Worked example

Find sin 60° cos 30° − sin 30° cos 60° exactly.

1. Substitute (√3/2)(√3/2) − (1/2)(1/2).
2. The first product is 3/4; the second is 1/4.
3. Subtract the fractions.

Answer: 1/2