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