Exact values for 30°, 45° and 60°
Halve an equilateral triangle and a square to find exact values of sine, cosine and tangent (trigonometric ratio) for 30°, 45° and 60°, link tan to the gradient of a line, and solve an equation such as sin x = 12.
- UK: Year 10
- France: grade 12
- Germany: grade 9
- Iran: grade 10
- Afghanistan: grade 9
- Syria: grade 9
- Palestine: grade 8
- Ukraine: grade 8
- Russia: grade 8
Guide
This studio is for Year 10 (GCSE G21: exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°). The Afghan books teach it in grade 9, the Iranian books in Year 10.
Cut an equilateral triangle of side 2 in half along its height. Each half is a right-angled triangle with angles 30°, 60° and 90°; its hypotenuse is 2, its short side 1 and its height, by Pythagoras' theorem, √(22 − 12) = √3. From this triangle: sin 30° = 12, cos 30° = √32, sin 60° = √32, cos 60° = 12.
The diagonal of a square of side 1 makes 45°, and the two sides next to it are equal, so tan 45° = 1 and sin 45° = cos 45° = √22. A line through the origin at angle α to the x-axis has gradient tanα.
Trigonometric ratios are not proportional to the angle: sin 60° is not twice sin 30°, and sin 30° is not 3090. Always read them from the triangle. Here we only use acute angles (between 0° and 90°).