Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).                     
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cos θ = ―1/2

Verified step by step guidance

1. Identify the trigonometric function and the given value: here, we have \( \cos \theta = -\frac{1}{2} \).
2. Recall that cosine corresponds to the x-coordinate on the unit circle, so we need to find all angles \( \theta \) in the interval \( [0^\circ, 360^\circ) \) where the x-coordinate is \( -\frac{1}{2} \).
3. Determine the reference angle: find the acute angle \( \alpha \) whose cosine is \( \frac{1}{2} \). This is the positive value, so \( \alpha = \cos^{-1} \left( \frac{1}{2} \right) \).
4. Since cosine is negative in the second and third quadrants, find the angles in these quadrants by using \( 180^\circ - \alpha \) and \( 180^\circ + \alpha \).
5. Write the solutions as \( \theta = 180^\circ - \alpha \) and \( \theta = 180^\circ + \alpha \), which are the angles where \( \cos \theta = -\frac{1}{2} \) within the given interval.

Key Concepts

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles on the unit circle can be measured in radians or degrees, and each point corresponds to (cos θ, sin θ). Understanding how to locate angles and their coordinates on the unit circle is essential for solving trigonometric equations.

Cosine Function and Its Values

The cosine of an angle θ corresponds to the x-coordinate of the point on the unit circle at that angle. Knowing the standard cosine values for common angles helps identify solutions where cos θ equals a specific value, such as -1/2, by finding all angles with that x-coordinate within the given interval.

Solving Trigonometric Equations over a Given Interval

When solving equations like cos θ = -1/2 over [0°, 360°) or [0, 2π), it is important to find all angles within the specified domain that satisfy the equation. This involves understanding the periodicity of trigonometric functions and identifying multiple solutions that correspond to the same cosine value.