In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator.tan⁻¹ (tan 2π/3)

Verified step by step guidance

1. Recognize that \( \tan^{-1} \) is the inverse function of \( \tan \), which means \( \tan^{-1}(\tan(x)) = x \) for \( x \) in the range of \( \tan^{-1} \).
2. The range of \( \tan^{-1} \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).
3. The given angle \( \frac{2\pi}{3} \) is not within this range, so we need to find an equivalent angle that is within the range.
4. Use the periodicity of the tangent function: \( \tan(x) = \tan(x + n\pi) \) for any integer \( n \).
5. Find an equivalent angle \( x \) such that \( \tan(x) = \tan\left(\frac{2\pi}{3}\right) \) and \( x \) is within \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan⁻¹, are used to find the angle whose tangent is a given number. They essentially reverse the action of the tangent function. For example, if y = tan(x), then x = tan⁻¹(y). The range of the inverse tangent function is limited to (-π/2, π/2), which is crucial for determining the correct angle when evaluating expressions.

Tangent Function and Its Periodicity

The tangent function, defined as the ratio of the sine and cosine functions (tan(x) = sin(x)/cos(x)), is periodic with a period of π. This means that tan(x) = tan(x + nπ) for any integer n. Understanding this periodicity is essential when evaluating expressions involving the tangent function, as it allows for simplification and finding equivalent angles.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They help in determining the values of trigonometric functions for angles greater than 90 degrees or less than 0 degrees. For example, the reference angle for 2π/3 is π/3, which is used to find the exact value of the tangent function in the context of the unit circle.