Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6.sec θ = -√2

Verified step by step guidance

1. Recognize that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, \( \sec \theta = -\sqrt{2} \) implies \( \cos \theta = -\frac{1}{\sqrt{2}} \).
2. Simplify \( \cos \theta = -\frac{1}{\sqrt{2}} \) to \( \cos \theta = -\frac{\sqrt{2}}{2} \).
3. Recall that \( \cos \theta = -\frac{\sqrt{2}}{2} \) corresponds to angles where the cosine value is negative, which occurs in the second and third quadrants.
4. Identify the reference angle where \( \cos \theta = \frac{\sqrt{2}}{2} \), which is \( 45^\circ \).
5. Determine the angles in the second and third quadrants: \( \theta = 180^\circ - 45^\circ \) and \( \theta = 180^\circ + 45^\circ \).

Key Concepts

Secant Function

The secant function, denoted as sec(θ), is the reciprocal of the cosine function. It is defined as sec(θ) = 1/cos(θ). Understanding the secant function is crucial for solving equations involving it, as it helps identify the relationship between secant and cosine values, particularly in determining the angles that yield specific secant values.

Quadrants and Sign of Trigonometric Functions

Trigonometric functions have different signs in different quadrants of the unit circle. In the case of sec(θ) = -√2, we need to identify in which quadrants the secant function is negative. Since sec(θ is negative in the second and third quadrants, this knowledge is essential for finding the correct angle solutions within the specified interval.

Finding Angles from Trigonometric Values

To find angles corresponding to a specific trigonometric value, one must use the inverse functions and consider the periodic nature of trigonometric functions. For sec(θ) = -√2, we first find the reference angle where sec(θ) = √2, then apply the appropriate quadrant adjustments to find all angles θ in the interval [0°, 360°) that satisfy the equation.