To determine the quadrant in which the terminal ray of the angle [tex]\(\frac{18\pi}{5}\)[/tex] radians lies, follow these steps:
1. Convert the given angle to a standard position:
- The angle provided is [tex]\(\frac{18\pi}{5}\)[/tex] radians.
2. Normalize the angle to fit within a full circle (0 to [tex]\(2\pi\)[/tex] radians):
- One full revolution is [tex]\(2\pi\)[/tex] radians.
- To normalize, we'll reduce our given angle modulo [tex]\(2\pi\)[/tex].
3. Compute [tex]\(\frac{18\pi}{5}\)[/tex] modulo [tex]\(2\pi\)[/tex]:
- [tex]\(\frac{18\pi}{5}\)[/tex] is approximately [tex]\(11.309733552923255\)[/tex] radians.
- When we normalize this angle, it results in about [tex]\(5.026548245743669\)[/tex] radians within the interval [tex]\([0, 2\pi)\)[/tex].
4. Determine the quadrant for the angle [tex]\(5.026548245743669\)[/tex] radians:
- Quadrants are divided as follows in radians:
- First Quadrant (0 to [tex]\(\frac{\pi}{2}\)[/tex]): [tex]\(0 \leq \theta < \frac{\pi}{2}\)[/tex]
- Second Quadrant ([tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex]): [tex]\(\frac{\pi}{2} \leq \theta < \pi\)[/tex]
- Third Quadrant ([tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex]): [tex]\(\pi \leq \theta < \frac{3\pi}{2}\)[/tex]
- Fourth Quadrant ([tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi\)[/tex]): [tex]\(\frac{3\pi}{2} \leq \theta < 2\pi\)[/tex]
- The interval where [tex]\(5.026548245743669\)[/tex] radians falls is [tex]\(\frac{3\pi}{2} \leq \theta < 2\pi\)[/tex].
- This places the angle in the Fourth Quadrant.
So, the terminal ray of the angle [tex]\(\frac{18\pi}{5}\)[/tex] radians lies in the Fourth Quadrant.
Thus, the correct answer is:
c IV
