Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. 1cos θ = - — 2

Verified step by step guidance

1. Recognize that \( \cos \theta = -\frac{1}{2} \) corresponds to an angle where the cosine value is negative.
2. Recall that cosine is negative in the second and third quadrants of the unit circle.
3. Identify the reference angle where \( \cos \theta = \frac{1}{2} \), which is \( 60^\circ \) or \( \pi/3 \) radians.
4. Determine the angles in the second quadrant: \( \theta = 180^\circ - 60^\circ \).
5. Determine the angles in the third quadrant: \( \theta = 180^\circ + 60^\circ \).

Key Concepts

Cosine Function

The cosine function, denoted as cos(θ), is a fundamental trigonometric function that relates the angle θ in a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is periodic with a range of values between -1 and 1, and its behavior is crucial for solving trigonometric equations.

Quadrants of the Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. The angles in standard position are measured from the positive x-axis, and the circle is divided into four quadrants, each with specific signs for sine and cosine. Understanding which quadrants correspond to positive or negative values of cosine is essential for finding all solutions to the equation.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccosine (cos⁻¹), are used to determine the angle corresponding to a given trigonometric ratio. For example, if cos(θ) = -1/2, we can use arccos to find the principal value of θ, but we must also consider all possible angles within the specified interval, which may include additional solutions based on the properties of the cosine function.