How to derive it
- Write 75° = 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 75°
| Ratio | Exact | Decimal |
|---|---|---|
| sin 75° | (√6 + √2)/4 | 0.9659 |
| cos 75° | (√6 − √2)/4 | 0.2588 |
| tan 75° | 2 + √3 | 3.7321 |
| cosec 75° | 1 ÷ sin 75° | 1.0353 |
| sec 75° | 1 ÷ cos 75° | 3.8637 |
| cot 75° | 1 ÷ tan 75° | 0.2679 |
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 75°?
2 + √3, which is about 3.7321.
What is tan 75° in radians?
75° is 5π/12 radians, so tan(5π/12) = 2 + √3.
Is tan 75° equal to cot 15°?
Yes. tan θ = cot(90° − θ), so tan 75° = cot 15°.
How do you find tan 75° without a calculator?
Write 75° = 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.