Evaluate each expression without using a calculator.
tan⁻¹ (tan (π/4))

Verified step by step guidance

1. Recognize that \( \tan^{-1} \) is the inverse function of \( \tan \), meaning \( \tan^{-1}(\tan(x)) = x \) for values of \( x \) within the range of \( \tan^{-1} \).
2. Identify the range of \( \tan^{-1} \), which is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).
3. Check if \( \frac{\pi}{4} \) is within the range of \( \tan^{-1} \).
4. Since \( \frac{\pi}{4} \) is within the range, \( \tan^{-1}(\tan(\frac{\pi}{4})) = \frac{\pi}{4} \).
5. Conclude that the expression evaluates to \( \frac{\pi}{4} \) because \( \tan^{-1} \) and \( \tan \) cancel each other out within the specified range.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan⁻¹ (arctangent), are used to find the angle whose tangent is a given number. They essentially reverse the action of the standard trigonometric functions. For example, tan⁻¹(tan(x)) will return x if x is within the principal range of the arctangent function, which is typically (-π/2, π/2).

Tangent Function

The tangent function, denoted as tan(x), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It is periodic with a period of π, meaning that tan(x) = tan(x + nπ) for any integer n. At specific angles, such as π/4, the tangent value is well-known, which simplifies calculations.

Principal Value

The principal value of an inverse trigonometric function is the unique output angle that lies within a specified range. For tan⁻¹, the principal value is restricted to the interval (-π/2, π/2). This means that when evaluating expressions like tan⁻¹(tan(π/4)), the result will be the angle within this range that corresponds to the tangent value, ensuring consistency in the output.