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To determine which set of values has the same reference angles, let's analyze each set and their reference angles.
### Set 1: [tex]\(\frac{\pi}{6}, \frac{\pi}{3}, \frac{5\pi}{6}\)[/tex]
- [tex]\(\frac{\pi}{6}\)[/tex]: The reference angle is [tex]\(\frac{\pi}{6}\)[/tex].
- [tex]\(\frac{\pi}{3}\)[/tex]: The reference angle is [tex]\(\frac{\pi}{3}\)[/tex].
- [tex]\(\frac{5\pi}{6}\)[/tex]: The reference angle is [tex]\(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\)[/tex].
This set has reference angles [tex]\(\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{6}\)[/tex].
### Set 2: [tex]\(\frac{\pi}{3}, \frac{5\pi}{6}, \frac{4\pi}{3}\)[/tex]
- [tex]\(\frac{\pi}{3}\)[/tex]: The reference angle is [tex]\(\frac{\pi}{3}\)[/tex].
- [tex]\(\frac{5\pi}{6}\)[/tex]: The reference angle is [tex]\(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\)[/tex].
- [tex]\(\frac{4\pi}{3}\)[/tex]: The reference angle is [tex]\(\frac{4\pi}{3} - \pi = \frac{\pi}{3}\)[/tex].
This set has reference angles [tex]\(\frac{\pi}{3}, \frac{\pi}{6}, \frac{\pi}{3}\)[/tex].
### Set 3: [tex]\(\frac{7}{2}, \frac{5\pi}{4}, \frac{7}{4}\)[/tex]
- [tex]\(\frac{7}{2}\)[/tex]: The reference angle is a bit unconventional due to not being a multiple of [tex]\(\pi\)[/tex] or straightforward fraction involving [tex]\(\pi\)[/tex].
- [tex]\(\frac{5\pi}{4}\)[/tex]: The reference angle is [tex]\(\pi + \frac{\pi}{4} = \frac{5\pi}{4} - \pi = \frac{\pi}{4}\)[/tex].
- [tex]\(\frac{7}{4}\)[/tex]: Again, this is unconventional without clear relation to [tex]\(\pi\)[/tex] in usual angle references.
The angles here do not clearly correspond to consistent reference angles due to uncommon forms.
### Set 4: [tex]\(\frac{\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4}\)[/tex]
- [tex]\(\frac{\pi}{4}\)[/tex]: The reference angle is [tex]\(\frac{\pi}{4}\)[/tex].
- [tex]\(\frac{3\pi}{4}\)[/tex]: The reference angle is [tex]\(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\)[/tex].
- [tex]\(\frac{7\pi}{4}\)[/tex]: The reference angle is [tex]\(2\pi - \frac{7\pi}{4} = \frac{\pi}{4}\)[/tex].
This set has reference angles [tex]\(\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{4}\)[/tex].
### Conclusion
From the detailed analysis of each set, it is evident that the values in the fourth set [tex]\(\frac{\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4}\)[/tex] all have the same reference angle of [tex]\(\frac{\pi}{4}\)[/tex].
Therefore, the set of values with the same reference angles is:
[tex]\[ \boxed{4} \][/tex]
### Set 1: [tex]\(\frac{\pi}{6}, \frac{\pi}{3}, \frac{5\pi}{6}\)[/tex]
- [tex]\(\frac{\pi}{6}\)[/tex]: The reference angle is [tex]\(\frac{\pi}{6}\)[/tex].
- [tex]\(\frac{\pi}{3}\)[/tex]: The reference angle is [tex]\(\frac{\pi}{3}\)[/tex].
- [tex]\(\frac{5\pi}{6}\)[/tex]: The reference angle is [tex]\(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\)[/tex].
This set has reference angles [tex]\(\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{6}\)[/tex].
### Set 2: [tex]\(\frac{\pi}{3}, \frac{5\pi}{6}, \frac{4\pi}{3}\)[/tex]
- [tex]\(\frac{\pi}{3}\)[/tex]: The reference angle is [tex]\(\frac{\pi}{3}\)[/tex].
- [tex]\(\frac{5\pi}{6}\)[/tex]: The reference angle is [tex]\(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\)[/tex].
- [tex]\(\frac{4\pi}{3}\)[/tex]: The reference angle is [tex]\(\frac{4\pi}{3} - \pi = \frac{\pi}{3}\)[/tex].
This set has reference angles [tex]\(\frac{\pi}{3}, \frac{\pi}{6}, \frac{\pi}{3}\)[/tex].
### Set 3: [tex]\(\frac{7}{2}, \frac{5\pi}{4}, \frac{7}{4}\)[/tex]
- [tex]\(\frac{7}{2}\)[/tex]: The reference angle is a bit unconventional due to not being a multiple of [tex]\(\pi\)[/tex] or straightforward fraction involving [tex]\(\pi\)[/tex].
- [tex]\(\frac{5\pi}{4}\)[/tex]: The reference angle is [tex]\(\pi + \frac{\pi}{4} = \frac{5\pi}{4} - \pi = \frac{\pi}{4}\)[/tex].
- [tex]\(\frac{7}{4}\)[/tex]: Again, this is unconventional without clear relation to [tex]\(\pi\)[/tex] in usual angle references.
The angles here do not clearly correspond to consistent reference angles due to uncommon forms.
### Set 4: [tex]\(\frac{\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4}\)[/tex]
- [tex]\(\frac{\pi}{4}\)[/tex]: The reference angle is [tex]\(\frac{\pi}{4}\)[/tex].
- [tex]\(\frac{3\pi}{4}\)[/tex]: The reference angle is [tex]\(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\)[/tex].
- [tex]\(\frac{7\pi}{4}\)[/tex]: The reference angle is [tex]\(2\pi - \frac{7\pi}{4} = \frac{\pi}{4}\)[/tex].
This set has reference angles [tex]\(\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{4}\)[/tex].
### Conclusion
From the detailed analysis of each set, it is evident that the values in the fourth set [tex]\(\frac{\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4}\)[/tex] all have the same reference angle of [tex]\(\frac{\pi}{4}\)[/tex].
Therefore, the set of values with the same reference angles is:
[tex]\[ \boxed{4} \][/tex]
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