tan15

tan15 is a mathematical expression representing the tangent of the angle 15 degrees. In trigonometry, the tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. For 15 degrees, this value can be derived using trigonometric identities, particularly those involving angle subtraction.

The exact value of tan(15°) can be calculated using the tangent of a difference formula:

tan(A − B) = (tan A − tan B) / (1 + tan A tan B)

For 15°, we can express it as 45° − 30°, where tan(45°) = 1 and tan(30°) = 1/√3. Substituting these values into the formula yields:

tan(15°) = (1 − 1/√3) / (1 + 1 * 1/√3)

= (√3 − 1) / (√3 + 1)

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator (√3 − 1):

tan(15°) = [(√3 − 1)(√3 − 1)] / [(√3 + 1)(√3 − 1)]

= (3 − 2√3 + 1) / (3 − 1)

= (4 − 2√3) / 2

= 2 − √3

Thus, the exact value of tan(15°) is 2 − √3, which is approximately 0.2679 when calculated numerically.

This value is useful in various mathematical and engineering applications, including trigonometric calculations, signal processing, and physics. It also appears in problems involving geometric constructions and exact trigonometric evaluations.