Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value.sec θ = -2√3 3

Verified step by step guidance

1. Recognize that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, \( \cos \theta = \frac{1}{\sec \theta} = -\frac{3}{2\sqrt{3}} \).
2. Simplify \( \cos \theta = -\frac{3}{2\sqrt{3}} \) by rationalizing the denominator: \( \cos \theta = -\frac{3\sqrt{3}}{6} = -\frac{\sqrt{3}}{2} \).
3. Identify the angles where \( \cos \theta = -\frac{\sqrt{3}}{2} \). These angles are in the second and third quadrants.
4. In the second quadrant, \( \theta = 180^\circ - 30^\circ = 150^\circ \).
5. In the third quadrant, \( \theta = 180^\circ + 30^\circ = 210^\circ \).

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 the given equation, as it helps identify the relationship between the angle θ and the cosine value needed to find the corresponding angle.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to specific signs of the trigonometric functions. In this case, since sec(θ) is negative, we need to determine in which quadrants the cosine function is also negative. This occurs in the second and third quadrants, which is essential for identifying the correct angles that satisfy the equation.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccosine, are used to find angles corresponding to specific trigonometric values. In this problem, once we determine the cosine value from the secant function, we can use the inverse cosine function to find the reference angle. This reference angle will then help us find all possible angles θ within the specified interval.