How to derive it
- Let θ = 18°, so 5θ = 90° and 2θ = 90° − 3θ.
- Then sin 2θ = cos 3θ: 2 sin θ cos θ = 4cos³θ − 3 cos θ.
- Divide by cos θ: 2 sin θ = 1 − 4 sin²θ, so 4 sin²θ + 2 sin θ − 1 = 0.
- Solve the quadratic and take the positive root: sin 18° = (√5 − 1)/4.
- cos²18° = 1 − sin²18° = 1 − (6 − 2√5)/16 = (10 + 2√5)/16.
- So cos 18° = √(10 + 2√5)/4.
All trig ratios of 18°
| Ratio | Exact | Decimal |
|---|---|---|
| sin 18° | (√5 − 1)/4 | 0.3090 |
| cos 18° | √(10 + 2√5)/4 | 0.9511 |
| tan 18° | √(25 − 10√5)/5 | 0.3249 |
| cosec 18° | 1 ÷ sin 18° | 3.2361 |
| sec 18° | 1 ÷ cos 18° | 1.0515 |
| cot 18° | 1 ÷ tan 18° | 3.0777 |
Exact values of special angles
For whole-degree values from 0° to 90°, see the trigonometry table.
Questions students ask
What is the value of cos 18°?
√(10 + 2√5)/4, which is about 0.9511.
What is cos 18° in radians?
18° is π/10 radians, so cos(π/10) = √(10 + 2√5)/4.
Is cos 18° equal to sin 72°?
Yes. cos θ = sin(90° − θ), so cos 18° = sin 72°.
How do you find cos 18° without a calculator?
Let θ = 18°, so 5θ = 90° and 2θ = 90° − 3θ. Then sin 2θ = cos 3θ: 2 sin θ cos θ = 4cos³θ − 3 cos θ. Divide by cos θ: 2 sin θ = 1 − 4 sin²θ, so 4 sin²θ + 2 sin θ − 1 = 0. Solve the quadratic and take the positive root: sin 18° = (√5 − 1)/4. cos²18° = 1 − sin²18° = 1 − (6 − 2√5)/16 = (10 + 2√5)/16. So cos 18° = √(10 + 2√5)/4.