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 sin 72° = √(10 + 2√5)/4.
All trig ratios of 72°
| Ratio | Exact | Decimal |
|---|---|---|
| sin 72° | √(10 + 2√5)/4 | 0.9511 |
| cos 72° | (√5 − 1)/4 | 0.3090 |
| tan 72° | √(5 + 2√5) | 3.0777 |
| cosec 72° | 1 ÷ sin 72° | 1.0515 |
| sec 72° | 1 ÷ cos 72° | 3.2361 |
| cot 72° | 1 ÷ tan 72° | 0.3249 |
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 sin 72°?
√(10 + 2√5)/4, which is about 0.9511.
What is sin 72° in radians?
72° is 2π/5 radians, so sin(2π/5) = √(10 + 2√5)/4.
Is sin 72° equal to cos 18°?
Yes. sin θ = cos(90° − θ), so sin 72° = cos 18°.
How do you find sin 72° 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 sin 72° = √(10 + 2√5)/4.