To convert polar coordinates to rectangular coordinates, we have a given polar coordinate [tex]\((r, \theta)\)[/tex]. The formulas to convert these to rectangular coordinates [tex]\((x, y)\)[/tex] are:
[tex]\[
x = r \cos(\theta)
\][/tex]
[tex]\[
y = r \sin(\theta)
\][/tex]
Given the polar coordinates are:
[tex]\[
r = 3 \sqrt{2}
\][/tex]
[tex]\[
\theta = \frac{3\pi}{4}
\][/tex]
We need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using the above formulas for [tex]\(r = 3 \sqrt{2}\)[/tex] and [tex]\(\theta = \frac{3\pi}{4}\)[/tex].
1. Calculate [tex]\(x\)[/tex]:
[tex]\[
x = r \cos(\theta)
\][/tex]
Substitute the values of [tex]\(r\)[/tex] and [tex]\(\theta\)[/tex]:
[tex]\[
x = 3 \sqrt{2} \cdot \cos\left(\frac{3\pi}{4}\right)
\][/tex]
We know the value of [tex]\(\cos\left(\frac{3\pi}{4}\right)\)[/tex] is [tex]\(-\frac{\sqrt{2}}{2}\)[/tex]:
[tex]\[
x = 3 \sqrt{2} \cdot -\frac{\sqrt{2}}{2}
\][/tex]
[tex]\[
x = 3 \sqrt{2} \cdot -\frac{\sqrt{2}}{2} = -3
\][/tex]
2. Calculate [tex]\(y\)[/tex]:
[tex]\[
y = r \sin(\theta)
\][/tex]
Substitute the values of [tex]\(r\)[/tex] and [tex]\(\theta\)[/tex]:
[tex]\[
y = 3 \sqrt{2} \cdot \sin\left(\frac{3\pi}{4}\right)
\][/tex]
We know the value of [tex]\(\sin\left(\frac{3\pi}{4}\right)\)[/tex] is [tex]\(\frac{\sqrt{2}}{2}\)[/tex]:
[tex]\[
y = 3 \sqrt{2} \cdot \frac{\sqrt{2}}{2}
\][/tex]
[tex]\[
y = 3 \sqrt{2} \cdot \frac{\sqrt{2}}{2} = 3
\][/tex]
Therefore, the corresponding rectangular coordinates are:
[tex]\[
(x, y) = (-3, 3)
\][/tex]
So, the correct answer is:
A. [tex]\((-3, 3)\)[/tex]
