Sin 15 Degrees Value: The exact value of sin 15° in fraction form is (√3 − 1) / (2√2) (or rationalised as (√6 − √2) / 4), and its approximate decimal value is 0.2588.
Value of Sin 15° in Fraction, Radical, and Decimal Forms
| Form | Value of sin 15° (or sin π/12) |
|---|---|
| Exact Radical Form 1 | (√3 − 1) / (2√2) |
| Exact Rationalised Form 2 | (√6 − √2) / 4 |
| Decimal Value (4 places) | 0.2588 (0.258819045…) |
| Angle in Radians | 15° = π / 12 radians (≈ 0.2618 rad) |
| Quadrant & Sign | First Quadrant (Positive) |
How to Find the Value of Sin 15°? (Step-by-Step Derivation)
We can derive the exact value of sin 15° easily using the compound angle subtraction formula sin(A − B) = sin A cos B − cos A sin B:
- Step 1: Express 15° using standard angles (45° and 30°):
Write15° = 45° − 30°(so A = 45° and B = 30°). - Step 2: Apply the trigonometric identity sin(A − B):
sin 15° = sin(45° − 30°) = sin 45° cos 30° − cos 45° sin 30° - Step 3: Substitute the standard values:
We know thatsin 45° = 1/√2,cos 30° = √3/2,cos 45° = 1/√2, andsin 30° = 1/2.sin 15° = (1/√2) × (√3/2) − (1/√2) × (1/2) - Step 4: Combine over the common denominator 2√2:
sin 15° = (√3 − 1) / (2√2) - Step 5: Rationalise the denominator (multiply numerator & denominator by √2):
sin 15° = [(√3 − 1) × √2] / (2√2 × √2) = (√6 − √2) / 4 - Step 6: Calculate the decimal value:
Substitute√6 ≈ 2.44949and√2 ≈ 1.41421:sin 15° = (2.44949 − 1.41421) / 4 = 1.03528 / 4 = 0.25882
Alternative Method: You can also write sin 15° = sin(60° − 45°) = sin 60° cos 45° − cos 60° sin 45° = (√3/2)(1/√2) − (1/2)(1/√2) = (√3 − 1)/(2√2), which gives the exact same value.
Related 15° and 75° Trigonometric Values Table
| Ratio | Exact Fraction Value | Approximate Decimal |
|---|---|---|
| sin 15° (= cos 75°) | (√3 − 1) / (2√2) = (√6 − √2) / 4 | 0.2588 |
| cos 15° (= sin 75°) | (√3 + 1) / (2√2) = (√6 + √2) / 4 | 0.9659 |
| tan 15° (= cot 75°) | (√3 − 1) / (√3 + 1) = 2 − √3 | 0.2679 |
| cot 15° (= tan 75°) | (√3 + 1) / (√3 − 1) = 2 + √3 | 3.7321 |
For full Class 11 Trigonometry chapters, see NCERT Class 11 Maths Book PDF.
Frequently Asked Questions on Sin 15 Degrees Value
What is the exact value of sin 15 degrees in fraction?
The exact fractional value of sin 15° is (√3 − 1)/(2√2). When rationalised by multiplying the numerator and denominator by √2, it is written as (√6 − √2)/4.
What is the decimal value of sin 15°?
The decimal value of sin 15 degrees is approximately 0.2588 (or 0.258819 when rounded to six decimal places).
Why is sin 15° equal to cos 75°?
By the complementary angle identity sin θ = cos(90° − θ), we have sin 15° = cos(90° − 15°) = cos 75° = (√6 − √2)/4.