Question
A point on the unit circle is \left((1)/(2), (√(3))/(2)\right). What is the sine of the angle formed by a terminal ray that intersects the unit circle at this point?
A. (1)/(2)
B. -(√(3))/(2)
C. (√(3))/(2)
D. -(1)/(2)
A. (1)/(2)
B. -(√(3))/(2)
C. (√(3))/(2)
D. -(1)/(2)
Answers
GPT-4o mini
The coordinates of the point on the unit circle are given as \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\). In the context of the unit circle, the x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle.
From the point \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\):
- The x-coordinate (cosine) is \(\cos(\theta) = \frac{1}{2}\).
- The y-coordinate (sine) is \(\sin(\theta) = \frac{\sqrt{3}}{2}\).
Therefore, the sine of the angle formed by the terminal ray that intersects the unit circle at this point is:
\[
\sin(\theta) = \frac{\sqrt{3}}{2}
\]
The correct answer is:
C. \(\frac{\sqrt{3}}{2}\)
From the point \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\):
- The x-coordinate (cosine) is \(\cos(\theta) = \frac{1}{2}\).
- The y-coordinate (sine) is \(\sin(\theta) = \frac{\sqrt{3}}{2}\).
Therefore, the sine of the angle formed by the terminal ray that intersects the unit circle at this point is:
\[
\sin(\theta) = \frac{\sqrt{3}}{2}
\]
The correct answer is:
C. \(\frac{\sqrt{3}}{2}\)