Which of the following explains why $\cos (60°)$ equals $\sin (30°)$ using the unit circle?

• A. On the unit circle, the x-coordinate at $60°$ is always greater than the y-coordinate at $30°$.
• B. Because $60°$ and $30°$ are both in the first quadrant, their cosine and sine values are always equal.
• C. The unit circle shows that $\cos (60°)$ and $\sin (30°)$ are both negative, so they must be equal.
• D. On the unit circle, the x-coordinate at $60°$ is equal to the y-coordinate at $30°$, so $\cos (60°)$ equals $\sin (30°)$.

Answer: (D)

Verified step by step guidance

1. Recall that on the unit circle, any point corresponding to an angle \( \theta \) has coordinates \( (\cos(\theta), \sin(\theta)) \). This means the x-coordinate is \( \cos(\theta) \) and the y-coordinate is \( \sin(\theta) \).
2. Identify the coordinates for the angles 60° and 30° on the unit circle: the point at 60° is \( (\cos(60^\circ), \sin(60^\circ)) \) and the point at 30° is \( (\cos(30^\circ), \sin(30^\circ)) \).
3. Notice that \( \cos(60^\circ) \) represents the x-coordinate of the point at 60°, and \( \sin(30^\circ) \) represents the y-coordinate of the point at 30°.
4. Using the symmetry of the unit circle, observe that the x-coordinate at 60° is the same as the y-coordinate at 30°, because 60° and 30° are complementary angles (they add up to 90°). This is a key property: \( \cos(\theta) = \sin(90^\circ - \theta) \).
5. Therefore, \( \cos(60^\circ) = \sin(30^\circ) \) because the x-coordinate at 60° equals the y-coordinate at 30° on the unit circle.