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. Recall that the function \( \tan^{-1}(x) \), also known as the arctangent, returns an angle \( \theta \) such that \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \). This is the principal value range for the inverse tangent function.
2. Identify the given expression: \( \tan^{-1}(\tan(\frac{2\pi}{3})) \). We want to find the angle in the principal range whose tangent is equal to \( \tan(\frac{2\pi}{3}) \).
3. Calculate or recall the value of \( \tan(\frac{2\pi}{3}) \). Since \( \frac{2\pi}{3} = 120^\circ \), which lies in the second quadrant, use the tangent identity: \( \tan(\pi - x) = -\tan(x) \). So, \( \tan(\frac{2\pi}{3}) = \tan(\pi - \frac{\pi}{3}) = -\tan(\frac{\pi}{3}) \).
4. Evaluate \( \tan(\frac{\pi}{3}) \), which is \( \sqrt{3} \), so \( \tan(\frac{2\pi}{3}) = -\sqrt{3} \).
5. Now, find \( \theta = \tan^{-1}(-\sqrt{3}) \) where \( \theta \) is in the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). Determine the angle \( \theta \) whose tangent is \( -\sqrt{3} \) within this principal range.

Key Concepts

Inverse Tangent Function (tan⁻¹ or arctan)

The inverse tangent function, arctan, returns the angle whose tangent is a given number. Its output range is limited to (-π/2, π/2), meaning it only returns angles in the first and fourth quadrants. Understanding this range is crucial when simplifying expressions involving arctan.

Periodicity and Properties of the Tangent Function

The tangent function has a period of π, so tan(θ) = tan(θ + nπ) for any integer n. This periodicity allows angles outside the principal range to be related to angles within it, which is essential when simplifying expressions like tan⁻¹(tan(2π/3)).

Simplifying Expressions Involving Inverse Trigonometric Functions

When evaluating expressions like tan⁻¹(tan(θ)), the result is the angle equivalent to θ but restricted to the principal range of arctan. This often requires adjusting θ by adding or subtracting multiples of π to find an equivalent angle within (-π/2, π/2).