Question

In which quadrant is an angle measuring start fraction 5 pi over 3 end fraction radians located, and what is its reference angle?
(1 point)
Responses

Quadrant III; start fraction pi over 6 end fraction
Quadrant III; Image with alt text: start fraction pi over 6 end fraction

Quadrant IV; start fraction pi over 6 end fraction
Quadrant IV; Image with alt text: start fraction pi over 6 end fraction

Quadrant III; start fraction pi over 3 end fraction
Quadrant III; Image with alt text: start fraction pi over 3 end fraction

Quadrant IV; start fraction pi over 3 end fraction

Answer 1

To determine the quadrant and reference angle for the angle measuring \( \frac{5\pi}{3} \) radians, follow these steps:

1. **Identify the quadrant**: The angle \( \frac{5\pi}{3} \) radians is greater than \( 2\pi \) (which is \( \frac{6\pi}{3} \)) but is also less than \( 2\pi \). To convert \( \frac{5\pi}{3} \) to degrees to visualize its position, you can use the conversion \( 180^\circ \) for \( \pi \) radians:
\[
\frac{5\pi}{3} \times \frac{180^\circ}{\pi} = 300^\circ
\]
The angle \( 300^\circ \) is in the fourth quadrant.

2. **Determine the reference angle**: The reference angle is obtained by finding the difference between the angle and the nearest x-axis (which is either \( 0 \) or \( 360^\circ \)):
\[
360^\circ - 300^\circ = 60^\circ
\]
To convert \( 60^\circ \) back to radians:
\[
60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3}
\]

Thus, for the angle \( \frac{5\pi}{3} \):
- It is located in **Quadrant IV**.
- The reference angle is \( \frac{\pi}{3} \).

Therefore, the correct response is:
**Quadrant IV; \( \frac{\pi}{3} \)**.