In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator.tan (tan⁻¹ 125)

Verified step by step guidance

1. Recognize that \( \tan^{-1} \) is the inverse function of \( \tan \), meaning \( \tan(\tan^{-1}(x)) = x \) for all \( x \) in the domain of \( \tan^{-1} \).
2. Identify the domain of \( \tan^{-1}(x) \), which is all real numbers, \( (-\infty, \infty) \).
3. Since 125 is a real number, it falls within the domain of \( \tan^{-1}(x) \).
4. Apply the property of inverse functions: \( \tan(\tan^{-1}(125)) = 125 \).
5. Conclude that the exact value of the expression \( \tan(\tan^{-1}(125)) \) is simply 125.

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. For example, tan⁻¹(125) gives the angle θ such that tan(θ) = 125. Understanding this concept is crucial for evaluating expressions involving inverse functions.

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 is also expressed as tan(θ) = sin(θ)/cos(θ). Recognizing how the tangent function operates is essential for simplifying expressions involving angles.

Composition of Functions

Composition of functions involves applying one function to the result of another. In this case, we are evaluating tan(tan⁻¹(125)), which simplifies to 125 because the tangent function and its inverse cancel each other out. Understanding function composition is key to solving problems that involve nested functions.