To find the rectangular coordinates (x, y) of a point given in polar form [tex]\((r, \theta)\)[/tex], we use the following equations:
[tex]\[
x = r \cos \theta
\][/tex]
[tex]\[
y = r \sin \theta
\][/tex]
Given the polar coordinates [tex]\( \left(-4, \frac{\pi}{3}\right) \)[/tex]:
1. Substitute [tex]\( r = -4 \)[/tex] and [tex]\( \theta = \frac{\pi}{3} \)[/tex] into the equations.
2. To find [tex]\( x \)[/tex]:
[tex]\[
x = -4 \cos \left(\frac{\pi}{3}\right)
\][/tex]
Using the trigonometric value [tex]\( \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \)[/tex]:
[tex]\[
x = -4 \left(\frac{1}{2}\right) = -2
\][/tex]
3. To find [tex]\( y \)[/tex]:
[tex]\[
y = -4 \sin \left(\frac{\pi}{3}\right)
\][/tex]
Using the trigonometric value [tex]\( \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \)[/tex]:
[tex]\[
y = -4 \left(\frac{\sqrt{3}}{2}\right) = -2\sqrt{3}
\][/tex]
So, the rectangular coordinates are [tex]\((-2, -2\sqrt{3})\)[/tex].
Given the multiple-choice options:
a. [tex]\((-2, -2 \sqrt{3})\)[/tex]
b. [tex]\((-2, 2 \sqrt{3})\)[/tex]
c. [tex]\( (2, -2 \sqrt{3})\)[/tex]
d. [tex]\( (2, 2 \sqrt{3})\)[/tex]
The correct answer is:
A
