Given [tex]\(\theta = \frac{9\pi}{4}\)[/tex], we aim to sketch this angle in standard position.
1. Identify the range: First, we need to determine the equivalent angle in the standard position, which must lie between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex].
2. Find the equivalent angle: Since [tex]\(\theta = \frac{9\pi}{4}\)[/tex] is more than [tex]\(2\pi\)[/tex], we need to reduce it within the range [tex]\(0 \leq \theta < 2\pi\)[/tex]:
- Subtract [tex]\(2\pi\)[/tex] from [tex]\(\frac{9\pi}{4}\)[/tex] to bring it into the desired range:
[tex]\[
\frac{9\pi}{4} - 2\pi = \frac{9\pi}{4} - \frac{8\pi}{4} = \frac{\pi}{4}
\][/tex]
3. Standard position: Thus, the equivalent angle is [tex]\(\frac{\pi}{4}\)[/tex].
4. Sketch the angle:
- Draw the x-axis and y-axis.
- Starting from the positive x-axis, measure an angle of [tex]\(\frac{\pi}{4}\)[/tex] in the counterclockwise direction.
- This angle, [tex]\(\frac{\pi}{4}\)[/tex], corresponds to 45 degrees.
- Draw a ray from the origin making an angle of [tex]\(\frac{\pi}{4}\)[/tex] with the positive x-axis.
The sketch will show a line originating from the origin and intersecting the first quadrant of the coordinate plane, making a 45-degree angle with the positive x-axis.
