To determine when \( \cos(\theta) = \frac{\sqrt{3}}{2} \) using special right triangles on the unit circle, we can start by recalling the properties of a 30-60-90 triangle.
In a 30-60-90 triangle:
- The sides are in the ratio \( 1 : \sqrt{3} : 2 \).
- The angle opposite the side measuring 1 (the shorter leg) is \( 30^\circ \).
- The angle opposite the side measuring \( \sqrt{3} \) (the longer leg) is \( 60^\circ \).
On the unit circle, the cosine of an angle corresponds to the x-coordinate of the point on the circle at that angle. Since cosine is positive in the first and fourth quadrants, we need to find the angles that yield \( \cos(\theta) = \frac{\sqrt{3}}{2} \).
- First Quadrant: The reference angle is \( 30^\circ \).
- Fourth Quadrant: The angle can be found using \( 360^\circ - 30^\circ = 330^\circ \).
Thus, the angles where \( \cos(\theta) = \frac{\sqrt{3}}{2} \) are:
- \( 30^\circ \)
- \( 330^\circ \)
In conclusion, for \( \cos(\theta) = \frac{\sqrt{3}}{2} \), the two angles in degrees are \( 30^\circ \) and \( 330^\circ \).