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. Recall that the function \( \sec^{-1}(x) \) is the inverse secant function, which gives the angle \( \theta \) such that \( \sec(\theta) = x \).
2. Understand that \( \sec(\theta) = \frac{1}{\cos(\theta)} \), so \( \sec^{-1}(-1) \) means finding \( \theta \) where \( \cos(\theta) = -1 \).
3. Identify the range of the inverse secant function, which is typically \( [0, \pi] \) excluding \( \frac{\pi}{2} \), to find the principal value of \( \theta \).
4. Determine the angle \( \theta \) within the range \( [0, \pi] \) where \( \cos(\theta) = -1 \).
5. Express the exact value of \( \sec^{-1}(-1) \) as the angle \( \theta \) found in the previous step.

Key Concepts

Inverse Secant Function (sec⁻¹)

The inverse secant function, sec⁻¹(x), returns the angle whose secant is x. Since secant is the reciprocal of cosine, sec⁻¹(x) = θ means sec(θ) = x, or cos(θ) = 1/x. Understanding the domain and range restrictions of sec⁻¹ is essential to find the correct angle.

Domain and Range of sec⁻¹

The domain of sec⁻¹(x) is |x| ≥ 1 because secant values are always ≤ -1 or ≥ 1. Its range is typically [0, π] excluding π/2, to ensure the function is one-to-one. Recognizing these restrictions helps identify valid angles for the inverse secant.

Exact Values of Trigonometric Functions

Finding exact values involves using known special angles and their trigonometric values without a calculator. For sec⁻¹(-1), knowing that sec(π) = -1 allows us to determine the exact angle. Familiarity with unit circle values is crucial for this process.