Question

Given the polar coordinates left-parenthesis 4 comma negative Start Fraction pi over 3 End Fraction right-parenthesis, which of the following rectangular coordinate pairs represents the same point?
(1 point)
Responses

left-parenthesis negative 2 comma 2 Start Root 3 End Root right-parenthesis
Image with alt text: left-parenthesis negative 2 comma 2 Start Root 3 End Root right-parenthesis

left-parenthesis 2 comma negative 2 Start Root 3 End Root right-parenthesis
Image with alt text: left-parenthesis 2 comma negative 2 Start Root 3 End Root right-parenthesis

left-parenthesis 2 Start Root 3 End Root comma negative 2 right-parenthesis
Image with alt text: left-parenthesis 2 Start Root 3 End Root comma negative 2 right-parenthesis

left-parenthesis 2 Start Root 3 End Root comma 2 right-parenthesis

Answer 1

To convert the polar coordinates \((4, -\frac{\pi}{3})\) to rectangular coordinates, we use the formulas:

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

where \(r\) is the radial coordinate (distance from the origin) and \(\theta\) is the angular coordinate (angle).

Given \(r = 4\) and \(\theta = -\frac{\pi}{3}\):

1. Calculate \(x\):
\[
x = 4 \cos\left(-\frac{\pi}{3}\right)
\]
The cosine of \(-\frac{\pi}{3}\) is equal to the cosine of \(\frac{\pi}{3}\) (since cosine is an even function):
\[
\cos\left(-\frac{\pi}{3}\right) = \frac{1}{2}
\]
Therefore:
\[
x = 4 \times \frac{1}{2} = 2
\]

2. Calculate \(y\):
\[
y = 4 \sin\left(-\frac{\pi}{3}\right)
\]
The sine of \(-\frac{\pi}{3}\) is the negative of the sine of \(\frac{\pi}{3}\):
\[
\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}
\]
Therefore:
\[
y = 4 \times -\frac{\sqrt{3}}{2} = -2\sqrt{3}
\]

Combining these results, the rectangular coordinates are:
\[
(2, -2\sqrt{3})
\]

Now we can find the matching answer from the given options:

- \((2, -2\sqrt{3})\) matches with the second option.

Thus, the answer is:
\[
\left(2, -2\sqrt{3}\right)
\]