In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator.tan⁻¹ [tan(− π/6)]

Verified step by step guidance

1. Recognize that \( \tan^{-1} \) and \( \tan \) are inverse functions, 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. Check if \( -\frac{\pi}{6} \) is within the range of \( \tan^{-1} \).
4. Since \( -\frac{\pi}{6} \) is within the range \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), \( \tan^{-1}(\tan(-\frac{\pi}{6})) = -\frac{\pi}{6} \).
5. Conclude that the exact value of the expression is \( -\frac{\pi}{6} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan⁻¹, are used to find the angle whose tangent is a given value. They essentially reverse the action of the trigonometric functions. 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.

Tangent Function and Its Periodicity

The tangent function, tan(x), is periodic with a period of π, meaning it repeats its values every π radians. This periodicity is important when evaluating expressions involving angles outside the standard range of the function. For instance, tan(−π/6) can be simplified using the periodicity of the tangent function to find equivalent angles within the principal range.

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 in different quadrants. For example, the reference angle for −π/6 is π/6, which allows us to evaluate tan(−π/6) using known values from the first quadrant.