use the special triangles on the unit circle to determine \\theta in degrees when \\sin \\theta =…

Explanation:

Step1: Recall the definition of sine in the unit circle

In the unit circle, for a point ((x,y)) on the terminal side of an angle (\theta), (\sin\theta=y).

Step2: Locate the (y -)coordinate

We are given (\sin\theta=\frac{\sqrt{3}}{2}). Looking at the points on the unit - circle diagram ((\frac{1}{2},\frac{\sqrt{3}}{2})) and ((\frac{\sqrt{3}}{2},\frac{1}{2})), the (y -)coordinate (\frac{\sqrt{3}}{2}) corresponds to the angle whose terminal side passes through the point ((\frac{1}{2},\frac{\sqrt{3}}{2})).

Step3: Identify the angle

The angle (\theta) whose terminal side passes through the point ((\frac{1}{2},\frac{\sqrt{3}}{2})) is (60^{\circ}) (since for the (30 - 60-90) triangle in the unit circle, when the side opposite the angle is (\frac{\sqrt{3}}{2}) (in the first - quadrant unit - circle context), the angle is (60^{\circ})).

Answer:

(60^{\circ})