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

Verified step by step guidance

1. Recognize that \( \frac{5\pi}{12} \) can be expressed as a sum of angles whose tangent values are known, such as \( \frac{\pi}{4} + \frac{\pi}{6} \).
2. Use the tangent addition formula: \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \).
3. Substitute \( a = \frac{\pi}{4} \) and \( b = \frac{\pi}{6} \) into the formula.
4. Calculate \( \tan \frac{\pi}{4} = 1 \) and \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \).
5. Substitute these values into the formula to find \( \tan \left( \frac{5\pi}{12} \right) = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} \).

Key Concepts

Tangent Function

The tangent function, denoted as tan(θ), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed in terms of sine and cosine as tan(θ) = sin(θ)/cos(θ). Understanding the properties and values of the tangent function is essential for solving trigonometric expressions.

Angle Addition Formula

The angle addition formula for tangent states that tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b)). This formula allows us to find the tangent of angles that are sums of known angles, which is particularly useful for calculating values like tan(5π/12) by expressing it as the sum of angles such as π/3 and π/4.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the specific values of sine, cosine, and tangent for commonly used angles, such as 0, π/6, π/4, π/3, and π/2. These values can be derived from the unit circle or special triangles. Knowing these exact values is crucial for simplifying expressions and solving trigonometric equations accurately.