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