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.
- cos 36° = 1 − 2 sin²18° = 1 − 2(6 − 2√5)/16 = (2 + 2√5)/8.
- So sin 54° = (√5 + 1)/4.
All trig ratios of 54°
| Ratio | Exact | Decimal |
|---|---|---|
| sin 54° | (√5 + 1)/4 | 0.8090 |
| cos 54° | √(10 − 2√5)/4 | 0.5878 |
| tan 54° | √(25 + 10√5)/5 | 1.3764 |
| cosec 54° | 1 ÷ sin 54° | 1.2361 |
| sec 54° | 1 ÷ cos 54° | 1.7013 |
| cot 54° | 1 ÷ tan 54° | 0.7265 |
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 54°?
(√5 + 1)/4, which is about 0.8090.
What is sin 54° in radians?
54° is 3π/10 radians, so sin(3π/10) = (√5 + 1)/4.
Is sin 54° equal to cos 36°?
Yes. sin θ = cos(90° − θ), so sin 54° = cos 36°.
How do you find sin 54° 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. cos 36° = 1 − 2 sin²18° = 1 − 2(6 − 2√5)/16 = (2 + 2√5)/8. So sin 54° = (√5 + 1)/4.