To determine if [tex]\(\frac{\pi}{4}\)[/tex] is the reference angle for any of the given angles, we need to calculate the reference angle for each one. The reference angle for an angle in radians is the smallest positive angle that is coterminal with it, or in other words, the angle obtained by reducing the given angle modulo [tex]\(2\pi\)[/tex].
Let's analyze each option in detail:
### A. [tex]\(\frac{15 \pi}{4}\)[/tex]
First, reduce [tex]\(\frac{15\pi}{4}\)[/tex] modulo [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{15\pi}{4} \mod 2\pi = \frac{15\pi}{4} \mod \frac{8\pi}{4} = \frac{15\pi}{4} \mod 2 = \frac{15\pi - 8\pi}{4} = \frac{7\pi}{4}
\][/tex]
Thus, the reference angle for [tex]\(\frac{15\pi}{4}\)[/tex] is [tex]\(\frac{7\pi}{4}\)[/tex].
### B. [tex]\(\frac{12 \pi}{4}\)[/tex]
Next, reduce [tex]\(\frac{12\pi}{4}\)[/tex] modulo [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{12\pi}{4} = 3\pi
\][/tex]
[tex]\[
3\pi \mod 2\pi = \pi
\][/tex]
Thus, the reference angle for [tex]\(\frac{12\pi}{4}\)[/tex] is [tex]\(\pi\)[/tex].
### C. [tex]\(\frac{19 \pi}{4}\)[/tex]
Next, reduce [tex]\(\frac{19\pi}{4}\)[/tex] modulo [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{19\pi}{4} \mod 2\pi = \frac{19\pi}{4} \mod \frac{8\pi}{4} = \frac{19\pi}{4} \mod 2 = \frac{19\pi - 16\pi}{4} = \frac{3\pi}{4}
\][/tex]
Thus, the reference angle for [tex]\(\frac{19\pi}{4}\)[/tex] is [tex]\(\frac{3\pi}{4}\)[/tex].
### D. [tex]\(\frac{7 \pi}{4}\)[/tex]
Finally, reduce [tex]\(\frac{7\pi}{4}\)[/tex] modulo [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{7\pi}{4} \mod 2\pi = \frac{7\pi}{4}
\][/tex]
Thus, the reference angle for [tex]\(\frac{7\pi}{4}\)[/tex] is itself [tex]\(\frac{7\pi}{4}\)[/tex].
Based on the above calculations, none of the given angles have [tex]\(\frac{\pi}{4}\)[/tex] as a reference angle. The reference angles computed are:
- [tex]\(\frac{15\pi}{4} \to \frac{7\pi}{4}\)[/tex]
- [tex]\(\frac{12\pi}{4} \to \pi\)[/tex]
- [tex]\(\frac{19\pi}{4} \to \frac{3\pi}{4}\)[/tex]
- [tex]\(\frac{7\pi}{4} \to \frac{7\pi}{4}\)[/tex]
Therefore, the correct answer is that [tex]\(\frac{\pi}{4}\)[/tex] is not the reference angle for any of the given options.
