How to derive it
- Write 15° = 45° − 30°.
- tan(A − B) = (tan A − tan B)/(1 + tan A tan B).
- = (1 − 1/√3)/(1 + 1/√3) = (√3 − 1)/(√3 + 1).
- Rationalise: (√3 − 1)² / 2 = (4 − 2√3)/2 = 2 − √3.
All trig ratios of 15°
| Ratio | Exact | Decimal |
|---|---|---|
| sin 15° | (√6 − √2)/4 | 0.2588 |
| cos 15° | (√6 + √2)/4 | 0.9659 |
| tan 15° | 2 − √3 | 0.2679 |
| cosec 15° | 1 ÷ sin 15° | 3.8637 |
| sec 15° | 1 ÷ cos 15° | 1.0353 |
| cot 15° | 1 ÷ tan 15° | 3.7321 |
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 15°?
2 − √3, which is about 0.2679.
What is tan 15° in radians?
15° is π/12 radians, so tan(π/12) = 2 − √3.
Is tan 15° equal to cot 75°?
Yes. tan θ = cot(90° − θ), so tan 15° = cot 75°.
How do you find tan 15° without a calculator?
Write 15° = 45° − 30°. tan(A − B) = (tan A − tan B)/(1 + tan A tan B). = (1 − 1/√3)/(1 + 1/√3) = (√3 − 1)/(√3 + 1). Rationalise: (√3 − 1)² / 2 = (4 − 2√3)/2 = 2 − √3.