Ratios of 0°, 30°, 45°, 60°, 90°

You don't need to learn this table by heart. Each value comes from a simple triangle.

45°: half a square. Cut a square of side a along its diagonal. You get a right triangle with two 45° angles, legs a and a, and hypotenuse √2 a. So sin 45° = cos 45° = 1/√2 and tan 45° = 1.

30° and 60°: half an equilateral triangle. Take an equilateral triangle of side 2a and cut it down the middle. You get a right triangle with sides a, √3 a and 2a, and angles 30°, 60° and 90°:

• sin 30° = a/2a = 1/2, cos 30° = √3/2, tan 30° = 1/√3
• sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

0° and 90°: shrink angle A towards 0°. The opposite side shrinks to nothing, so sin 0° = 0 and cos 0° = 1. Grow A towards 90° and it's the other way round: sin 90° = 1, cos 90° = 0.

The pattern: the cos row is the sin row backwards. As A goes from 0° to 90°, sin rises from 0 to 1 and cos falls from 1 to 0. And tan = sin ÷ cos.

'Not defined' is not a number, and not 'infinity'. tan 90° = 1/0, and you can't divide by 0. The same goes for sec 90°, cot 0° and cosec 0°. But cot 90° = cos 90°/sin 90° = 0/1 = 0, which is fine.

Finding an angle: get the ratio alone, then match it to the table. If sin θ = 1/2, then θ = 30°. Keep the number that belongs to the angle: 2cos 3θ = 1 gives cos 3θ = 1/2, so 3θ = 60° and θ = 20°.

sin²θ means (sin θ)²: work out sin θ first, then square it.