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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sin x = ―1/2

Verified step by step guidance

1. Identify the trigonometric function and the given value: here, \( \sin x = -\frac{1}{2} \). We need to find all angles \( x \) in the interval \( [0, 2\pi) \) where the sine value is \( -\frac{1}{2} \).
2. Recall that sine corresponds to the y-coordinate on the unit circle. Since \( \sin x = -\frac{1}{2} \), we are looking for points on the unit circle where the y-coordinate is \( -\frac{1}{2} \).
3. Determine the reference angle: find the angle in the first quadrant whose sine is \( \frac{1}{2} \). This reference angle is \( \frac{\pi}{6} \) because \( \sin \frac{\pi}{6} = \frac{1}{2} \).
4. Since sine is negative in the third and fourth quadrants, find the angles in these quadrants by adding the reference angle to \( \pi \) and subtracting it from \( 2\pi \), respectively. These give the solutions \( x = \pi + \frac{\pi}{6} \) and \( x = 2\pi - \frac{\pi}{6} \).
5. Write the final solution set as \( \left\{ \pi + \frac{\pi}{6}, 2\pi - \frac{\pi}{6} \right\} \), which are the values of \( x \) in \( [0, 2\pi) \) where \( \sin x = -\frac{1}{2} \).

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 are measured in radians (0 to 2π) or degrees (0° to 360°), and each angle corresponds to a point (x, y) where x = cos(θ) and y = sin(θ). Understanding this helps identify the angle(s) where sine takes a specific value.

Sine Function Values on the Unit Circle

The sine of an angle corresponds to the y-coordinate of the point on the unit circle at that angle. To solve sin x = -1/2, one must find all angles in the given interval where the y-coordinate equals -1/2, considering the sine function’s periodicity and symmetry.

Solving Trigonometric Equations in a Given Interval

When solving equations like sin x = -1/2 over [0, 2π) or [0°, 360°), it is essential to find all solutions within the specified domain. This involves identifying reference angles and using the unit circle’s symmetry to determine all valid angles where the equation holds true.