Answer :
Let's determine which set of polar coordinates corresponds to the same point as the given point [tex]\(\left(5, \frac{\pi}{4}\right)\)[/tex].
First, convert the given polar coordinates [tex]\(\left(5, \frac{\pi}{4}\right)\)[/tex] to Cartesian coordinates to understand the point in the [tex]\(xy\)[/tex]-plane:
[tex]\[ x = r \cos(\theta) = 5 \cos\left(\frac{\pi}{4}\right) = 5 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \][/tex]
[tex]\[ y = r \sin(\theta) = 5 \sin\left(\frac{\pi}{4}\right) = 5 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \][/tex]
So the Cartesian coordinates for the given polar coordinates [tex]\(\left(5, \frac{\pi}{4}\right)\)[/tex] are [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex].
Now, let's check each option by converting them to Cartesian coordinates and see which one matches [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex]:
### Option A: [tex]\(\left(5, \frac{3\pi}{4}\right)\)[/tex]
[tex]\[ x = 5 \cos\left(\frac{3\pi}{4}\right) = 5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\frac{5\sqrt{2}}{2} \][/tex]
[tex]\[ y = 5 \sin\left(\frac{3\pi}{4}\right) = 5 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \][/tex]
This gives Cartesian coordinates [tex]\(\left(-\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex], which does not match [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex].
### Option B: [tex]\(\left(5, -\frac{\pi}{4}\right)\)[/tex]
[tex]\[ x = 5 \cos\left(-\frac{\pi}{4}\right) = 5 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \][/tex]
[tex]\[ y = 5 \sin\left(-\frac{\pi}{4}\right) = 5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\frac{5\sqrt{2}}{2} \][/tex]
This gives Cartesian coordinates [tex]\(\left(\frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2}\right)\)[/tex], which does not match [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex].
### Option C: [tex]\(\left(-5, \frac{5\pi}{4}\right)\)[/tex]
[tex]\[ x = -5 \cos\left(\frac{5\pi}{4}\right) = -5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = \frac{5\sqrt{2}}{2} \][/tex]
[tex]\[ y = -5 \sin\left(\frac{5\pi}{4}\right) = -5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = \frac{5\sqrt{2}}{2} \][/tex]
This gives Cartesian coordinates [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex], which matches [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex].
### Option D: [tex]\(\left(-5, \frac{7\pi}{4}\right)\)[/tex]
[tex]\[ x = -5 \cos\left(\frac{7\pi}{4}\right) = -5 \cdot \frac{\sqrt{2}}{2} = -\frac{5\sqrt{2}}{2} \][/tex]
[tex]\[ y = -5 \sin\left(\frac{7\pi}{4}\right) = -5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = \frac{5\sqrt{2}}{2} \][/tex]
This gives Cartesian coordinates [tex]\(\left(-\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex], which does not match [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex].
Comparing all results, we see that only Option C [tex]\(\left(-5, \frac{5\pi}{4}\right)\)[/tex] gives the same point as [tex]\(\left(5, \frac{\pi}{4}\right)\)[/tex] in Cartesian coordinates.
Therefore, the correct option is [tex]\(\boxed{C}\)[/tex].
First, convert the given polar coordinates [tex]\(\left(5, \frac{\pi}{4}\right)\)[/tex] to Cartesian coordinates to understand the point in the [tex]\(xy\)[/tex]-plane:
[tex]\[ x = r \cos(\theta) = 5 \cos\left(\frac{\pi}{4}\right) = 5 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \][/tex]
[tex]\[ y = r \sin(\theta) = 5 \sin\left(\frac{\pi}{4}\right) = 5 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \][/tex]
So the Cartesian coordinates for the given polar coordinates [tex]\(\left(5, \frac{\pi}{4}\right)\)[/tex] are [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex].
Now, let's check each option by converting them to Cartesian coordinates and see which one matches [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex]:
### Option A: [tex]\(\left(5, \frac{3\pi}{4}\right)\)[/tex]
[tex]\[ x = 5 \cos\left(\frac{3\pi}{4}\right) = 5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\frac{5\sqrt{2}}{2} \][/tex]
[tex]\[ y = 5 \sin\left(\frac{3\pi}{4}\right) = 5 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \][/tex]
This gives Cartesian coordinates [tex]\(\left(-\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex], which does not match [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex].
### Option B: [tex]\(\left(5, -\frac{\pi}{4}\right)\)[/tex]
[tex]\[ x = 5 \cos\left(-\frac{\pi}{4}\right) = 5 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \][/tex]
[tex]\[ y = 5 \sin\left(-\frac{\pi}{4}\right) = 5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\frac{5\sqrt{2}}{2} \][/tex]
This gives Cartesian coordinates [tex]\(\left(\frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2}\right)\)[/tex], which does not match [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex].
### Option C: [tex]\(\left(-5, \frac{5\pi}{4}\right)\)[/tex]
[tex]\[ x = -5 \cos\left(\frac{5\pi}{4}\right) = -5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = \frac{5\sqrt{2}}{2} \][/tex]
[tex]\[ y = -5 \sin\left(\frac{5\pi}{4}\right) = -5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = \frac{5\sqrt{2}}{2} \][/tex]
This gives Cartesian coordinates [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex], which matches [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex].
### Option D: [tex]\(\left(-5, \frac{7\pi}{4}\right)\)[/tex]
[tex]\[ x = -5 \cos\left(\frac{7\pi}{4}\right) = -5 \cdot \frac{\sqrt{2}}{2} = -\frac{5\sqrt{2}}{2} \][/tex]
[tex]\[ y = -5 \sin\left(\frac{7\pi}{4}\right) = -5 \cdot \left(-\frac{\sqrt{2}}{2}\right) = \frac{5\sqrt{2}}{2} \][/tex]
This gives Cartesian coordinates [tex]\(\left(-\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex], which does not match [tex]\(\left(\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2}\right)\)[/tex].
Comparing all results, we see that only Option C [tex]\(\left(-5, \frac{5\pi}{4}\right)\)[/tex] gives the same point as [tex]\(\left(5, \frac{\pi}{4}\right)\)[/tex] in Cartesian coordinates.
Therefore, the correct option is [tex]\(\boxed{C}\)[/tex].
