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The unit circle is a concept that helps visualize and understand trigonometric functions. Imagine a circle with a radius of 1, centered at the origin of a coordinate plane. This circle is special because:

• Each point on its circumference corresponds to a unique angle formed with the positive x-axis.
• The x-coordinate of these points gives you the cosine of the angle, while the y-coordinate gives you the sine.
• The unit circle allows for the representation of all angles from 0° to 360° or 0 to 2Ï€ radians.

In our exercise, the value of \( \sin \theta = \frac{1}{2} \) corresponds to specific points on the unit circle, one of which is in the second quadrant. This helps us locate which specific angle \( \theta \) is being referred to by looking at the location where the circle's y-coordinate equals \( \frac{1}{2} \). Notably, understanding the unit circle is crucial when determining solutions for trigonometric equations, especially when specifying angles within air-bound ranges like 0° to 360°.

A reference angle is a helpful concept in trigonometry that refers to the smallest angle between the terminal side of the given angle and the x-axis. It is always measured in the positive direction.

• Reference angles are always acute, meaning they are between 0° and 90°.
• They are used to find the other angles that have the same trigonometric function outputs.
• For example, if \( \sin \theta = \frac{1}{2} \), the reference angle is \(30^{\circ}\) because \( \sin 30^{\circ} = \frac{1}{2} \).

In our step-by-step solution, knowing \( 30^{\circ} \) as the reference angle helps us identify potential angles within specific quadrants that match the required sine value. It's this reference angle that guides us to 150° in the second quadrant since the behavior of sine in the second quadrant aligns with that of the reference angle.

The coordinate plane is divided into four quadrants, and these are essential for understanding trigonometric functions and their signs:

• Quadrant I: Both sine and cosine are positive. Angles here range from 0° to 90°.
• Quadrant II: Sine is positive, cosine is negative. Angles here range from 90° to 180°.
• Quadrant III: Both sine and cosine are negative. Angles here range from 180° to 270°.
• Quadrant IV: Sine is negative, cosine is positive. Angles here range from 270° to 360°.

In our context, because the given angle terminates in Quadrant II, we deduce that sine is positive while cosine is negative. This information, combined with the reference angle, tells us the specific angle we are looking for is 150°. By understanding these quadrant properties, we can quickly assess the sign and values of trigonometric functions across different sections of the unit circle.