Question

Given the polar coordinates left-parenthesis negative 2 comma Start Fraction 3 pi over 2 End Fraction right-parenthesis, which of the following rectangular coordinate pairs represents the same point?
(1 point)
Responses

left-parenthesis 0 comma negative 2 right-parenthesis
Image with alt text: left-parenthesis 0 comma negative 2 right-parenthesis

left-parenthesis 2 comma 0 right-parenthesis
Image with alt text: left-parenthesis 2 comma 0 right-parenthesis

left-parenthesis 0 comma 2 right-parenthesis
Image with alt text: left-parenthesis 0 comma 2 right-parenthesis

left-parenthesis negative 2 comma 0 right-parenthesis

Answer 1

To convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), we use the formulas:

\[
x = r \cos(\theta)
\]
\[
y = r \sin(\theta)
\]

Given the polar coordinates \((-2, \frac{3\pi}{2})\):

1. **Identify \(r\) and \(\theta\)**:
- \(r = -2\)
- \(\theta = \frac{3\pi}{2}\)

2. **Calculate \(x\) and \(y\)**:
- First compute \(\cos(\frac{3\pi}{2})\) and \(\sin(\frac{3\pi}{2})\):
- \(\cos(\frac{3\pi}{2}) = 0\)
- \(\sin(\frac{3\pi}{2}) = -1\)

3. **Substitute into the equations**:
\[
x = -2 \cos\left(\frac{3\pi}{2}\right) = -2 \cdot 0 = 0
\]
\[
y = -2 \sin\left(\frac{3\pi}{2}\right) = -2 \cdot -1 = 2
\]

Thus, the rectangular coordinates are \((0, 2)\).

The correct answer is:
\[
\left(0, 2\right)
\]