Find the polar coordinates [tex]\((r, \theta)\)[/tex], where [tex]\(\theta\)[/tex] is in degrees, for the rectangular coordinates [tex]\((2, -2)\)[/tex].

A. [tex]\(\left(2 \sqrt{2}, 45^{\circ}\right)\)[/tex]
B. [tex]\(\left(-2 \sqrt{2}, -45^{\circ}\right)\)[/tex]
C. [tex]\(\left(2 \sqrt{2}, 315^{\circ}\right)\)[/tex]
D. [tex]\(\left(\sqrt{2}, -45^{\circ}\right)\)[/tex]



Answer :

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]