To convert the rectangular coordinates [tex]\((5, -5)\)[/tex] to polar coordinates [tex]\((r, \theta)\)[/tex], we need to follow these steps:
1. Compute the radius [tex]\( r \)[/tex]:
The radius [tex]\( r \)[/tex] is the distance from the origin to the point [tex]\((x, y)\)[/tex] and is given by the formula:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
Substituting [tex]\( x = 5 \)[/tex] and [tex]\( y = -5 \)[/tex], we get:
[tex]\[
r = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}
\][/tex]
2. Compute the angle [tex]\( \theta \)[/tex]:
The angle [tex]\( \theta \)[/tex] is the angle formed with the positive x-axis and is given by:
[tex]\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\][/tex]
Substituting [tex]\( x = 5 \)[/tex] and [tex]\( y = -5 \)[/tex], we get:
[tex]\[
\theta = \tan^{-1}\left(\frac{-5}{5}\right) = \tan^{-1}(-1)
\][/tex]
The arctangent of [tex]\(-1\)[/tex] is [tex]\(-\frac{\pi}{4}\)[/tex]. However, since the point [tex]\((5, -5)\)[/tex] is in the fourth quadrant, we need to add [tex]\(2\pi\)[/tex] to [tex]\(-\frac{\pi}{4}\)[/tex] to get a positive angle:
[tex]\[
\theta = 2\pi - \frac{\pi}{4} = \frac{8\pi}{4} - \frac{\pi}{4} = \frac{7\pi}{4}
\][/tex]
So, the polar coordinates for the point [tex]\((5, -5)\)[/tex] are:
[tex]\[
\boxed{\left(5\sqrt{2}, \frac{7\pi}{4}\right)}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\left(5\sqrt{2}, \frac{7\pi}{4}\right)
\][/tex]
