How to derive it
- Write 75° = 45° + 30°.
- cos(A + B) = cos A cos B − sin A sin B.
- = (1/√2)(√3/2) − (1/√2)(1/2) = (√3 − 1)/(2√2).
- Multiply top and bottom by √2: (√6 − √2)/4.
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 cos 75°?
(√6 − √2)/4, which is about 0.2588.
What is cos 75° in radians?
75° is 5π/12 radians, so cos(5π/12) = (√6 − √2)/4.
Is cos 75° equal to sin 15°?
Yes. cos θ = sin(90° − θ), so cos 75° = sin 15°.
How do you find cos 75° without a calculator?
Write 75° = 45° + 30°. cos(A + B) = cos A cos B − sin A sin B. = (1/√2)(√3/2) − (1/√2)(1/2) = (√3 − 1)/(2√2). Multiply top and bottom by √2: (√6 − √2)/4.