Find the exact value of each real number y if it exists. Do not use a calculator.
y = sec⁻¹ 1

Verified step by step guidance

1. Understand that \( y = \sec^{-1}(1) \) means we are looking for an angle \( y \) such that \( \sec(y) = 1 \).
2. Recall that the secant function, \( \sec(y) \), is the reciprocal of the cosine function, \( \cos(y) \). Therefore, \( \sec(y) = 1 \) implies \( \cos(y) = 1 \).
3. Identify the angle \( y \) for which \( \cos(y) = 1 \). This occurs at \( y = 0 \) radians (or \( y = 0^\circ \)).
4. Verify that \( y = 0 \) is within the range of the inverse secant function, which is \([0, \pi] \) excluding \( \frac{\pi}{2} \).
5. Conclude that the exact value of \( y \) is \( 0 \) radians, as it satisfies all conditions.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sec⁻¹ (arcsec), are used to find the angle whose secant is a given number. For example, if y = sec⁻¹(1), we are looking for an angle θ such that sec(θ) = 1. Understanding these functions is crucial for solving problems involving angles and their corresponding trigonometric ratios.

Secant Function

The secant function, denoted as sec(θ), is defined as the reciprocal of the cosine function: sec(θ) = 1/cos(θ). It is important to know the values of secant at key angles, as this helps in determining the angles corresponding to specific secant values, such as sec(0) = 1, which is relevant for finding y in the given problem.

Range and Domain of Inverse Functions

The range of the inverse secant function, sec⁻¹(x), is restricted to angles in the intervals [0, π/2) and (π/2, π]. This means that when solving for y = sec⁻¹(1), we must consider these intervals to find valid angle solutions. Understanding the domain and range of inverse functions is essential for correctly interpreting their outputs.