In Exercises 1–26, find the exact value of each expression. _ sec⁻¹ (−√2)

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. Identify the value inside the inverse secant: \( x = -\sqrt{2} \). We want to find \( \theta \) such that \( \sec(\theta) = -\sqrt{2} \).
3. Recall the relationship between secant and cosine: \( \sec(\theta) = \frac{1}{\cos(\theta)} \). So, \( \frac{1}{\cos(\theta)} = -\sqrt{2} \) implies \( \cos(\theta) = -\frac{1}{\sqrt{2}} \).
4. Determine the angle(s) \( \theta \) in the principal range of \( \sec^{-1} \), which is usually \( [0, \pi] \) excluding \( \frac{\pi}{2} \), where \( \cos(\theta) = -\frac{1}{\sqrt{2}} \).
5. Identify the exact angle(s) where cosine equals \( -\frac{1}{\sqrt{2}} \), which correspond to known special angles, and select the one within the principal range of the inverse secant function.

Key Concepts

Inverse Secant Function (sec⁻¹)

The inverse secant function, sec⁻¹(x), returns the angle whose secant is x. It is defined for |x| ≥ 1, and its range is typically [0, π] excluding π/2. Understanding this function helps find the angle corresponding to a given secant value.

Secant and Cosine Relationship

Secant is the reciprocal of cosine, so sec(θ) = 1/cos(θ). To find sec⁻¹(x), you can think in terms of cosine: cos(θ) = 1/x. This relationship is crucial for converting between secant and cosine values when solving trigonometric equations.

Evaluating Exact Values Using Special Angles

Exact values of trigonometric functions often come from special angles like π/4, π/3, and π/6. Recognizing that sec(3π/4) = -√2, for example, allows you to determine the exact angle for sec⁻¹(−√2) without a calculator.