Concept 8.2
Ratios of 30°, 45° and 60°
Source: NCERT Mathematics, Class 1010 minute lesson
Formal statement
sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2, tan 45° = 1; sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.
The intuition
You don't have to memorise this table. Cut an equilateral triangle of side 2 in half: you get angles 30°, 60°, 90° and sides 1, √3, 2.
Read every 30° and 60° value off that half-triangle. The side opposite 30° is the short one, 1, so sin 30° = 1/2.
The attention
Look at the half-triangle: the short side is opposite 30°.
Half of an equilateral triangle: 1 opposite 30°, √3 opposite 60°, 2 hypotenuse
Look hereNot here
- Look here
- The short side, 1. The shortest side faces the smallest angle. So 30° faces 1, giving sin 30° = 1/2 and tan 30° = 1/√3.
- Not here
- The height, √3. √3 is opposite 60°, not 30°. Mixing them swaps sin 30° and sin 60°, the commonest table mistake.
- What it teaches
- Rebuild a fact from a picture instead of memorising it.