To determine the rectangular coordinates for a point given in polar coordinates, we use the following formulas:
1. [tex]\( x = r \cdot \cos(\theta) \)[/tex]
2. [tex]\( y = r \cdot \sin(\theta) \)[/tex]
In this problem, the polar coordinates are [tex]\( (r, \theta) = (4, \pi) \)[/tex]. We need to find the corresponding rectangular coordinates [tex]\((x, y)\)[/tex].
1. Calculate [tex]\( x \)[/tex]:
[tex]\[
x = 4 \cdot \cos(\pi)
\][/tex]
Based on the properties of the cosine function:
[tex]\[
\cos(\pi) = -1
\][/tex]
So,
[tex]\[
x = 4 \cdot -1 = -4
\][/tex]
2. Calculate [tex]\( y \)[/tex]:
[tex]\[
y = 4 \cdot \sin(\pi)
\][/tex]
Based on the properties of the sine function:
[tex]\[
\sin(\pi) = 0
\][/tex]
So,
[tex]\[
y = 4 \cdot 0 = 0
\][/tex]
Hence, the rectangular coordinates corresponding to the polar coordinates [tex]\( (4, \pi) \)[/tex] are [tex]\( (-4, 0) \)[/tex].
Now, let’s check the given options to identify the coordinate that matches [tex]\( (-4, 0) \)[/tex]:
A. [tex]\( (4,0) \)[/tex]
B. [tex]\( (-4,0) \)[/tex] ← This matches our calculated coordinates.
C. [tex]\( (0,-4) \)[/tex]
D. [tex]\( (0,4) \)[/tex]
Therefore, the correct answer is B. [tex]$(-4,0)$[/tex]~
