Find the exact value of each expression.
tan (5π/12)

Verified step by step guidance

1. Recognize that the angle \( \frac{5\pi}{12} \) radians is not one of the standard angles with known tangent values, so we need to express it as a sum or difference of angles whose tangent values we know. For example, \( \frac{5\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4} \).
2. Recall the tangent addition formula: \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]. We will use this formula with \( A = \frac{\pi}{3} \) and \( B = \frac{\pi}{4} \).
3. Substitute the known tangent values: \( \tan \frac{\pi}{3} = \sqrt{3} \) and \( \tan \frac{\pi}{4} = 1 \) into the formula: \[ \tan \left( \frac{\pi}{3} + \frac{\pi}{4} \right) = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \times 1} \].
4. Simplify the numerator and denominator separately to get the expression \( \frac{\sqrt{3} + 1}{1 - \sqrt{3}} \).
5. To find the exact value, rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator \( (1 + \sqrt{3}) \), then simplify the resulting expression.

Key Concepts

Angle Sum and Difference Identities

These identities allow the calculation of trigonometric functions for angles expressed as sums or differences of known angles. For tangent, the formula is tan(a ± b) = (tan a ± tan b) / (1 ∓ tan a tan b). This is essential for finding tan(5π/12) by breaking it into angles with known tangent values.

Exact Values of Common Angles

Certain angles like π/4, π/3, and π/6 have well-known exact trigonometric values. Recognizing these angles helps in decomposing complex angles into sums or differences of these standard angles, enabling exact value calculations without a calculator.

Radian Measure

Radian is a unit of angular measure based on the radius of a circle. Understanding how to convert and interpret angles in radians is crucial, as trigonometric functions often use radian inputs, and the problem involves an angle expressed in radians (5π/12).