The value of tan 15° = 2 − √3 ≈ 0.2679. It is derived using the subtraction formula for tangent: tan(A − B) = (tan A − tan B)/(1 + tan A · tan B). Setting A = 45° and B = 30° gives tan 15° = tan(45° − 30°) = (1 − 1/√3)/(1 + 1/√3) = 2 − √3.
tan 15° = 2 − √3 ≈ 0.2679.
Derived using tan(45° − 30°) = (tan 45° − tan 30°)/(1 + tan 45° · tan 30°) = (1 − 1/√3)/(1 + 1/√3).
After rationalisation: tan 15° = (√3−1)²/2 = (4−2√3)/2 = 2−√3.
tan 75° = 2 + √3 ≈ 3.7321 (complement of tan 15°).
tan 15° × tan 75° = (2−√3)(2+√3) = 1.
sin 15° = (√6−√2)/4 ≈ 0.2588; cos 15° = (√6+√2)/4 ≈ 0.9659.
15° = 45° − 30° = 60° − 45°; both decompositions yield tan 15° = 2 − √3.
tan 15° = 2 − √3
Numerical value: √3 ≈ 1.7321 tan 15° = 2 − 1.7321 = 0.2679
Verification: tan 15° ≈ 0.26795 (from calculator) ✓
tan 15° is positive because 15° is in the first quadrant (0° to 90°), where all trigonometric functions are positive.
Note: 15° = 45° − 30° = 60° − 45°. Both decompositions can be used to derive the value.
The tangent subtraction formula is: tan(A − B) = (tan A − tan B) / (1 + tan A · tan B)
Set A = 45° and B = 30°, so A − B = 15°.
Known values:
Substitute into the formula: tan 15° = (tan 45° − tan 30°) / (1 + tan 45° · tan 30°) = (1 − 1/√3) / (1 + 1 · 1/√3) = (1 − 1/√3) / (1 + 1/√3)
Multiply numerator and denominator by √3: = (√3 − 1) / (√3 + 1)
Rationalise by multiplying by (√3 − 1)/(√3 − 1): = (√3 − 1)² / [(√3 + 1)(√3 − 1)] = (3 − 2√3 + 1) / (3 − 1) = (4 − 2√3) / 2 = 2 − √3
Therefore: tan 15° = 2 − √3 ≈ 0.2679 ✓
We can also compute tan 15° as tan(60° − 45°):
tan(A − B) = (tan A − tan B) / (1 + tan A · tan B)
Set A = 60°, B = 45°:
tan 15° = (√3 − 1) / (1 + √3 · 1) = (√3 − 1) / (√3 + 1)
Rationalise: = (√3 − 1)(√3 − 1) / [(√3 + 1)(√3 − 1)] = (√3 − 1)² / (3 − 1) = (3 − 2√3 + 1) / 2 = (4 − 2√3) / 2 = 2 − √3
Same result: tan 15° = 2 − √3 ✓
Both methods give the same answer, confirming the result.
Using the reference angle 15° (in the first quadrant, all values positive):
Starting from the known values for 45° and 30°:
| Function | Exact Value | Decimal |
|---|---|---|
| sin 15° | (√6 − √2)/4 | 0.2588 |
| cos 15° | (√6 + √2)/4 | 0.9659 |
| tan 15° | 2 − √3 | 0.2679 |
| cot 15° | 2 + √3 | 3.7321 |
| sec 15° | √6 − √2 | 1.0353 |
| cosec 15° | √6 + √2 | 3.8637 |
Key relationships:
Verification of sin²15° + cos²15°: [(√6−√2)/4]² + [(√6+√2)/4]² = [(6−2√12+2) + (6+2√12+2)] / 16 = [8−2√12+8+2√12] / 16 = 16/16 = 1 ✓
tan 15° and tan 75° are complementary values: tan 75° = cot 15° = 2 + √3
Proof that 1/(2−√3) = 2+√3: Multiply numerator and denominator by (2+√3): 1/(2−√3) × (2+√3)/(2+√3) = (2+√3)/(4−3) = (2+√3)/1 = 2+√3 ✓
Derivation of tan 75°: tan 75° = tan(45°+30°) = (tan 45° + tan 30°) / (1 − tan 45° · tan 30°) = (1 + 1/√3) / (1 − 1/√3) = (√3 + 1) / (√3 − 1) = (√3 + 1)² / (3 − 1) = (4 + 2√3) / 2 = 2 + √3
Summary:
tan 15° = 2 − √3 ≈ 0.2679. It is derived using the tangent subtraction formula: tan(45° − 30°) = (1 − 1/√3)/(1 + 1/√3) = (√3 − 1)/(√3 + 1) = 2 − √3 after rationalisation.
Using tan(A−B) = (tan A − tan B)/(1 + tan A · tan B), set A = 45°, B = 30°: tan 15° = (1 − 1/√3)/(1 + 1/√3) = (√3−1)/(√3+1). Multiply by (√3−1)/(√3−1): = (√3−1)²/2 = (4−2√3)/2 = 2−√3.
cot 15° = 1/tan 15° = 1/(2−√3). Rationalising: 1/(2−√3) × (2+√3)/(2+√3) = (2+√3)/(4−3) = 2+√3 ≈ 3.7321.
tan 75° = cot 15° = 2 + √3 ≈ 3.7321. It can be derived using tan(45°+30°) = (1 + 1/√3)/(1 − 1/√3) = (√3+1)/(√3−1) = (4+2√3)/2 = 2+√3.
sin 15° = (√6 − √2)/4 ≈ 0.2588 and cos 15° = (√6 + √2)/4 ≈ 0.9659. These are derived using sin(45°−30°) and cos(45°−30°) formulas.
Diagonal of a Cuboid — Space Diagonal Formula
Diagonal of a cuboid = √(l²+b²+h²). Space diagonal is the longest diagonal inside a cuboid. Formula derivation, face diagonals, and solved examples.
Properties of the Diagonals of a Rhombus
Learn the three most important geometric properties of the diagonals of a rhombus: they bisect each other, they meet at 90 degrees, and they bisect the angles.
Differentiation of sin⁻¹(x)
Learn the formula for the differentiation of sin inverse x. See the step-by-step derivation using the chain rule for Class 12 Calculus.
What is the Directly Proportional Symbol? ($\propto$)
Learn the meaning of the directly proportional symbol (∝). Understand how to type it, its mathematical meaning, and the constant of proportionality (k).
What is the Formula for Distance?
Learn the universal mathematical formula for Distance. Understand how to calculate distance using Speed and Time, with simple real-world train examples.
Turn this guide into revision flashcards, a practice exam, or an AI-generated podcast — free, no signup required.