How to derive it
- Use the half-angle formula with θ = 135°: cos²(θ/2) = (1 + cos θ)/2.
- cos 135° = −1/√2, so cos²67.5° = (1 − 1/√2)/2 = (2 − √2)/4.
- cos 67.5° is positive, so cos 67.5° = √(2 − √2)/2.
All trig ratios of 67.5°
| Ratio | Exact | Decimal |
|---|---|---|
| sin 67.5° | √(2 + √2)/2 | 0.9239 |
| cos 67.5° | √(2 − √2)/2 | 0.3827 |
| tan 67.5° | √2 + 1 | 2.4142 |
| cosec 67.5° | 1 ÷ sin 67.5° | 1.0824 |
| sec 67.5° | 1 ÷ cos 67.5° | 2.6131 |
| cot 67.5° | 1 ÷ tan 67.5° | 0.4142 |
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 cos 67.5°?
√(2 − √2)/2, which is about 0.3827.
What is cos 67.5° in radians?
67.5° is 3π/8 radians, so cos(3π/8) = √(2 − √2)/2.
Is cos 67.5° equal to sin 22.5°?
Yes. cos θ = sin(90° − θ), so cos 67.5° = sin 22.5°.
How do you find cos 67.5° without a calculator?
Use the half-angle formula with θ = 135°: cos²(θ/2) = (1 + cos θ)/2. cos 135° = −1/√2, so cos²67.5° = (1 − 1/√2)/2 = (2 − √2)/4. cos 67.5° is positive, so cos 67.5° = √(2 − √2)/2.