How to derive it
- Use the half-angle formula with θ = 45°: sin²(θ/2) = (1 − cos θ)/2.
- cos 45° = 1/√2, so sin²22.5° = (1 − 1/√2)/2 = (2 − √2)/4.
- sin 22.5° is positive, so sin 22.5° = √(2 − √2)/2.
All trig ratios of 22.5°
| Ratio | Exact | Decimal |
|---|---|---|
| sin 22.5° | √(2 − √2)/2 | 0.3827 |
| cos 22.5° | √(2 + √2)/2 | 0.9239 |
| tan 22.5° | √2 − 1 | 0.4142 |
| cosec 22.5° | 1 ÷ sin 22.5° | 2.6131 |
| sec 22.5° | 1 ÷ cos 22.5° | 1.0824 |
| cot 22.5° | 1 ÷ tan 22.5° | 2.4142 |
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 22.5°?
√(2 − √2)/2, which is about 0.3827.
What is sin 22.5° in radians?
22.5° is π/8 radians, so sin(π/8) = √(2 − √2)/2.
Is sin 22.5° equal to cos 67.5°?
Yes. sin θ = cos(90° − θ), so sin 22.5° = cos 67.5°.
How do you find sin 22.5° without a calculator?
Use the half-angle formula with θ = 45°: sin²(θ/2) = (1 − cos θ)/2. cos 45° = 1/√2, so sin²22.5° = (1 − 1/√2)/2 = (2 − √2)/4. sin 22.5° is positive, so sin 22.5° = √(2 − √2)/2.