To find the polar coordinates [tex]\((r, \theta)\)[/tex] for the given rectangular coordinates [tex]\((2, -2)\)[/tex], we will go through a series of steps.
### Step-by-Step Solution:
1. Find the Radius [tex]\( r \)[/tex]:
The radius [tex]\( r \)[/tex] is the distance from the origin to the point [tex]\((x, y)\)[/tex]. It can be calculated using the Pythagorean theorem:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
Here, [tex]\( x = 2 \)[/tex] and [tex]\( y = -2 \)[/tex]:
[tex]\[
r = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\][/tex]
2. Find the Angle [tex]\( \theta \)[/tex] in Radians:
The angle [tex]\( \theta \)[/tex] can be found using the arctangent function:
[tex]\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\][/tex]
Here, [tex]\( x = 2 \)[/tex] and [tex]\( y = -2 \)[/tex]:
[tex]\[
\theta = \tan^{-1}\left(\frac{-2}{2}\right) = \tan^{-1}(-1)
\][/tex]
The arctangent of [tex]\(-1\)[/tex] yields [tex]\(-\frac{\pi}{4}\)[/tex] radians because arctangent gives angles in the range [tex]\(-\frac{\pi}{2}\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex].
3. Convert the Angle to Degrees:
To convert from radians to degrees, we use the conversion factor [tex]\(\frac{180}{\pi}\)[/tex]:
[tex]\[
\theta = -\frac{\pi}{4} \times \frac{180}{\pi} = -45^\circ
\][/tex]
4. Adjust the Angle to be Between 0 and 360 Degrees:
Since we generally prefer angles in polar coordinates to be between 0 and 360 degrees:
[tex]\[
\theta = -45^\circ + 360^\circ = 315^\circ
\][/tex]
So the polar coordinates [tex]\((r, \theta)\)[/tex] are:
[tex]\[
(2\sqrt{2}, 315^\circ)
\][/tex]
Among the given options, the correct answer is:
[tex]\[
\left(2 \sqrt{2}, 315^\circ\right)
\][/tex]
