How to derive it
- Use tan(θ/2) = (1 − cos θ)/sin θ with θ = 135°.
- = (1 + 1/√2)/(1/√2) = √2 + 1.
- So tan 67.5° = √2 + 1.
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 tan 67.5°?
√2 + 1, which is about 2.4142.
What is tan 67.5° in radians?
67.5° is 3π/8 radians, so tan(3π/8) = √2 + 1.
Is tan 67.5° equal to cot 22.5°?
Yes. tan θ = cot(90° − θ), so tan 67.5° = cot 22.5°.
How do you find tan 67.5° without a calculator?
Use tan(θ/2) = (1 − cos θ)/sin θ with θ = 135°. = (1 + 1/√2)/(1/√2) = √2 + 1. So tan 67.5° = √2 + 1.