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

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1. Understand that \( \sec^{-1}(x) \) represents the angle whose secant is \( x \).
2. Recall that the secant function, \( \sec(\theta) \), is the reciprocal of the cosine function: \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
3. Set \( \sec(\theta) = -\sqrt{2} \), which implies \( \cos(\theta) = -\frac{1}{\sqrt{2}} \).
4. Recognize that \( \cos(\theta) = -\frac{1}{\sqrt{2}} \) corresponds to the angle \( \theta = \frac{3\pi}{4} \) or \( \theta = \frac{5\pi}{4} \) in the unit circle, where cosine is negative.
5. Choose the principal value for \( \sec^{-1} \), which is in the range \([0, \pi] \), so the angle is \( \theta = \frac{3\pi}{4} \).

Key Concepts

Inverse Secant Function

The inverse secant function, denoted as sec⁻¹(x), is the function that returns the angle whose secant is x. It is defined for x values outside the interval (-1, 1), specifically for x ≤ -1 or x ≥ 1. Understanding this function is crucial for solving problems involving secant and its inverses.

Understanding Negative Values

When dealing with sec⁻¹(−√2), it is important to recognize that the negative value indicates that we are looking for an angle in the second or third quadrant, where the secant (1/cosine) is negative. This understanding helps in determining the correct angle that corresponds to the given secant value.

Exact Values of Trigonometric Functions

Finding exact values of trigonometric functions often involves using special angles, such as 30°, 45°, and 60°. For sec⁻¹(−√2), knowing that sec(135°) = -√2 allows us to conclude that the exact value of the expression is 135° or 3π/4 radians, which is essential for providing a precise answer.