To draw the angle [tex]\(\frac{23 \pi}{8}\)[/tex] on the unit circle, we'll consider the following steps:
### Step-by-Step Solution:
1. Understanding the Angle in Radians:
The angle given is [tex]\(\frac{23 \pi}{8}\)[/tex]. This angle is in radians.
2. Convert the Angle to Degrees:
To convert the angle from radians to degrees, we use the conversion factor [tex]\(\frac{180^\circ}{\pi}\)[/tex]:
[tex]\[
\frac{23 \pi}{8} \times \frac{180^\circ}{\pi} = \frac{23 \times 180^\circ}{8} = 517.5^\circ
\][/tex]
3. Normalize the Angle:
Since the angles on the unit circle are usually considered within the range of [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex], we need to find the equivalent angle within this range. To do this, we subtract [tex]\(360^\circ\)[/tex] until the angle falls into this range:
[tex]\[
517.5^\circ - 360^\circ = 157.5^\circ
\][/tex]
So, [tex]\(\frac{23 \pi}{8}\)[/tex] radians corresponds to an angle of [tex]\(157.5^\circ\)[/tex].
4. Draw the Angle:
To draw [tex]\(157.5^\circ\)[/tex] on the unit circle:
- Start from the positive x-axis.
- Rotate counterclockwise through [tex]\(157.5^\circ\)[/tex].
[tex]\(157.5^\circ\)[/tex] is located in the second quadrant.
5. Determine Coordinates on the Unit Circle:
The coordinates of an angle [tex]\(\theta\)[/tex] on the unit circle are [tex]\((\cos \theta, \sin \theta)\)[/tex]. For [tex]\(\theta = \frac{23 \pi}{8}\)[/tex]:
- Calculate Cosine:
[tex]\[
\cos(\frac{23 \pi}{8}) = -0.9238795325112864
\][/tex]
- Calculate Sine:
[tex]\[
\sin(\frac{23 \pi}{8}) = 0.3826834323650905
\][/tex]
6. Plot the Point:
The point corresponding to [tex]\(\frac{23 \pi}{8}\)[/tex] radians on the unit circle is [tex]\((-0.9238795325112864, 0.3826834323650905)\)[/tex].
Thus, the point [tex]\((-0.9238795325112864, 0.3826834323650905)\)[/tex] lies on the unit circle at an angle of [tex]\(\frac{23 \pi}{8}\)[/tex] radians, which corresponds to [tex]\(157.5^\circ\)[/tex] counterclockwise from the positive x-axis.
