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°).
<IMAGE>
cos x = 1/2

Verified step by step guidance

1. Recall that on the unit circle, the cosine of an angle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle.
2. Identify the angles in the interval \([0, 2\pi)\) where the cosine value is \(\frac{1}{2}\). These are the angles where the x-coordinate equals \(\frac{1}{2}\).
3. From the unit circle, note that \(\cos x = \frac{1}{2}\) at two standard positions: one in the first quadrant and one in the fourth quadrant.
4. Write down the general solutions for \(x\) in radians: \(x = \frac{\pi}{3}\) (first quadrant) and \(x = 2\pi - \frac{\pi}{3}\) (fourth quadrant).
5. Verify that both solutions lie within the interval \([0, 2\pi)\) and list them as the final answers.

Key Concepts

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. It helps relate angles to coordinates, where the x-coordinate corresponds to cosine and the y-coordinate to sine of the angle. Angles can be measured in radians (0 to 2π) or degrees (0° to 360°), and understanding this is essential for solving trigonometric equations within specified intervals.

Cosine Function and Its Values

The cosine of an angle in the unit circle is the x-coordinate of the corresponding point. Knowing the cosine values for common angles (like π/3 or 60° where cos = 1/2) allows you to identify solutions to equations such as cos x = 1/2. Since cosine is positive in the first and fourth quadrants, multiple solutions may exist within the given interval.

Solving Trigonometric Equations Over an Interval

Solving equations like cos x = 1/2 over a specific interval requires finding all angles within that range that satisfy the equation. This involves identifying reference angles and considering the symmetry of the unit circle, as cosine values repeat in different quadrants. Solutions must be expressed within the given domain, either [0, 2π) for radians or [0°, 360°) for degrees.