Evaluate each expression without using a calculator.
sec (sec⁻¹ 2)

Verified step by step guidance

1. Understand that \( \sec^{-1}(x) \) is the inverse function of \( \sec(x) \), meaning \( \sec(\sec^{-1}(x)) = x \) for values of \( x \) within the domain of \( \sec^{-1} \).
2. Recognize that the domain of \( \sec^{-1}(x) \) is \( |x| \geq 1 \), and the range is \( [0, \pi] \) excluding \( \frac{\pi}{2} \).
3. Given \( \sec^{-1}(2) \), identify that \( 2 \) is within the domain of \( \sec^{-1} \).
4. Apply the property of inverse functions: \( \sec(\sec^{-1}(2)) = 2 \).
5. Conclude that the expression evaluates to \( 2 \) without further calculation.

Key Concepts

Secant Function

The secant function, denoted as sec(x), is the reciprocal of the cosine function. It is defined as sec(x) = 1/cos(x). The secant function is important in trigonometry as it relates to the angles of a right triangle and can be used to find lengths of sides when the angle is known.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sec⁻¹(x), are used to find the angle whose secant is a given value. For example, sec⁻¹(2) gives the angle θ such that sec(θ) = 2. Understanding these functions is crucial for solving problems that involve finding angles from trigonometric ratios.

Composition of Functions

The composition of functions involves applying one function to the result of another. In this case, evaluating sec(sec⁻¹(2)) means finding the secant of the angle whose secant is 2. This concept is essential for simplifying expressions involving inverse functions and understanding their relationships.