In Exercises 29–51, find the exact value of each expression. Do not use a calculator.sec⁻¹ (−1)

Verified step by step guidance

1. Understand that \( \sec^{-1}(x) \) is the inverse secant function, which gives the angle whose secant is \( x \).
2. Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Therefore, \( \sec^{-1}(-1) \) means we are looking for an angle \( \theta \) such that \( \sec(\theta) = -1 \).
3. Since \( \sec(\theta) = \frac{1}{\cos(\theta)} \), we have \( \frac{1}{\cos(\theta)} = -1 \), which implies \( \cos(\theta) = -1 \).
4. Identify the angle \( \theta \) where \( \cos(\theta) = -1 \). This occurs at \( \theta = \pi \) (or 180 degrees) within the range of the inverse secant function, which is \([0, \pi]\) excluding \( \frac{\pi}{2} \).
5. Conclude that the exact value of \( \sec^{-1}(-1) \) is \( \pi \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sec⁻¹ (also known as arcsec), are used to find the angle whose secant is a given value. The secant function is defined as the reciprocal of the cosine function, so understanding how to manipulate these relationships is crucial for solving problems involving inverse secant.

Domain and Range of Secant

The secant function has specific domain and range restrictions that are important when finding its inverse. The secant function is defined for all real numbers except where the cosine is zero, and its range is limited to values greater than or equal to 1 or less than or equal to -1. This affects the values that can be input into the inverse function.

Exact Values of Trigonometric Functions

Finding exact values of trigonometric functions often involves using special angles (like 0, 30, 45, 60, and 90 degrees) and their corresponding sine, cosine, and tangent values. For sec⁻¹ (−1), recognizing that the secant of an angle is -1 at specific angles (like 120° or 240°) is essential for determining the exact value without a calculator.