Let's solve for the equivalent angle, its quadrant, and the reference angle for [tex]\(\frac{16 \pi}{9}\)[/tex].
1. Finding the Equivalent Angle Between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex]:
The given angle is [tex]\(\theta = \frac{16 \pi}{9}\)[/tex].
To find the equivalent angle, we need to subtract multiples of [tex]\(2\pi\)[/tex] (because [tex]\(\theta = \theta + 2\pi k\)[/tex] for any integer [tex]\(k\)[/tex]).
[tex]\[
\text{Equivalent}\ \theta = \frac{16 \pi}{9} \mod 2 \pi
\][/tex]
The equivalent angle (taking modulus) is approximately [tex]\(5.585\)[/tex] radians.
2. Determine the Quadrant of the Equivalent Angle:
- [tex]\(0 \leq \theta < \frac{\pi}{2}\)[/tex]: Quadrant 1
- [tex]\(\frac{\pi}{2} \leq \theta < \pi\)[/tex]: Quadrant 2
- [tex]\(\pi \leq \theta < \frac{3\pi}{2}\)[/tex]: Quadrant 3
- [tex]\(\frac{3\pi}{2} \leq \theta < 2\pi\)[/tex]: Quadrant 4
Since [tex]\(5.585 \; \text{radians}\)[/tex] is between [tex]\(\frac{3\pi}{2}\)[/tex] (which is approximately 4.712) and [tex]\(2\pi\)[/tex] (which is approximately 6.283), the angle is in Quadrant 4.
3. Calculate the Reference Angle:
The reference angle [tex]\(\bar{\theta}\)[/tex] is the acute angle formed with the closest x-axis.
For angles in Quadrant 4, the reference angle [tex]\(\bar{\theta}\)[/tex] can be found by:
[tex]\[
\bar{\theta} = 2\pi - \theta
\][/tex]
Substituting our equivalent angle:
[tex]\[
\bar{\theta} = 2\pi - 5.585
\][/tex]
The reference angle is approximately [tex]\(0.698 \; \text{radians}\)[/tex].
Summarizing this information in our table, we get:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Angle } \theta & \text{Quadrant} & \text{Reference Angle } \bar{\theta} \\
\hline
135^{\circ} & 2 & 45^{\circ} \\
\hline
255^{\circ} & 3 & 75^{\circ} \\
\hline
\frac{\pi}{3} & 1 & 60^{\circ} \\
\hline
\frac{16\pi}{9} & 4 & 0.698 \; \text{radians} \\
\hline
\end{array}
\][/tex]
The angle [tex]\(\frac{16 \pi}{9}\)[/tex] is in Quadrant 4 with a reference angle of approximately 0.698 radians.
