To determine the two pairs of polar coordinates for the Cartesian point [tex]\((-3, 0)\)[/tex], we'll follow a structured approach:
1. Convert Cartesian coordinates to polar coordinates:
The polar coordinates [tex]\((r, \theta)\)[/tex] can be expressed through the formulas:
- [tex]\( r = \sqrt{x^2 + y^2} \)[/tex]
- [tex]\( \theta = \tan^{-1}\left(\frac{y}{x}\right) \)[/tex] (adjusting for the quadrant)
For the point [tex]\((-3, 0)\)[/tex]:
- [tex]\( x = -3 \)[/tex]
- [tex]\( y = 0 \)[/tex]
So,
- [tex]\( r = \sqrt{(-3)^2 + 0^2} = \sqrt{9} = 3 \)[/tex]
- [tex]\( \theta = \tan^{-1}\left(\frac{0}{-3}\right) = \tan^{-1}(0) = 0^{\circ} \)[/tex]
However, since [tex]\(x\)[/tex] is negative and [tex]\(y\)[/tex] is zero, the angle must actually be [tex]\(180^{\circ}\)[/tex]. Therefore, the coordinates are:
- [tex]\((r, \theta) = (3, 180^{\circ})\)[/tex]
2. Find the second pair of polar coordinates:
In polar coordinates, the negative [tex]\(r\)[/tex] with an adjusted [tex]\(\theta\)[/tex] also describes the same point.
- The second pair involves negating the radius,
[tex]\( r_2 = -r = -3 \)[/tex]
- and adding [tex]\(180^{\circ}\)[/tex] to [tex]\(\theta\)[/tex],
[tex]\( \theta_2 = 180^{\circ} + 180^{\circ} = 360^{\circ} \)[/tex]. Since angles in polar coordinates are typically represented within [tex]\(0^{\circ} \leq \theta < 360^{\circ}\)[/tex], we adjust this angle:
[tex]\( \theta_2 = 360^{\circ} - 360^{\circ} = 0^{\circ} \)[/tex]
Thus, the second pair of polar coordinates is:
- [tex]\((r_2, \theta_2) = (-3, 0^{\circ})\)[/tex]
Combining both findings, the two pairs of polar coordinates for [tex]\((-3, 0)\)[/tex] are:
- [tex]\((3, 180^{\circ})\)[/tex]
- [tex]\((-3, 0^{\circ})\)[/tex]
Therefore, the best answer from the choices provided is:
A. [tex]\(\left(3,180^{\circ}\right),\left(-3,0^{\circ}\right)\)[/tex]
