To determine which of the given angle measures corresponds to an angle measuring [tex]\(\frac{\pi}{8}\)[/tex] radians, we need to convert the angle from radians to degrees and then round our answer to two decimal places.
Here is a detailed, step-by-step solution:
1. Understanding the Conversion Factor:
To convert an angle from radians to degrees, we use the conversion factor [tex]\(180^{\circ}/\pi\)[/tex]:
[tex]\[
\text{degrees} = \text{radians} \times \frac{180^\circ}{\pi}
\][/tex]
2. Substitute the Given Value:
We are given [tex]\(\frac{\pi}{8}\)[/tex] radians. Substitute this value into the conversion formula:
[tex]\[
\text{degrees} = \frac{\pi}{8} \times \frac{180^\circ}{\pi}
\][/tex]
3. Simplify the Expression:
The [tex]\(\pi\)[/tex] terms cancel out, and the calculation simplifies to:
[tex]\[
\text{degrees} = \frac{180^\circ}{8} = 22.5^\circ
\][/tex]
4. Rounding:
The answer [tex]\(22.5^\circ\)[/tex] is already rounded to two decimal places, so no further rounding is needed.
Now we compare this result with the provided options:
- Option A: [tex]\(22^\circ 5'\)[/tex] (which is [tex]\(22.083^\circ\)[/tex], not a match)
- Option B: [tex]\(39.25^\circ\)[/tex] (not a match)
- Option C: [tex]\(22^\circ 30'\)[/tex] (which is [tex]\(22.5^\circ\)[/tex], a match)
- Option D: [tex]\(22.5^\circ\)[/tex] (a match)
Therefore, the correct answers are C and D:
- [tex]\(22^\circ 30'\)[/tex]
- [tex]\(22.5^\circ\)[/tex]
