To determine two pairs of polar coordinates for the point [tex]\((3, -3)\)[/tex] when [tex]\(0^{\circ} < \theta < 360^{\circ}\)[/tex], we need to follow a few steps.
### Step 1: Convert Cartesian Coordinates to Polar Coordinates
First, we need to convert the Cartesian coordinates [tex]\((3, -3)\)[/tex] into polar form [tex]\((r, \theta)\)[/tex]. The polar coordinates are determined as follows:
1. Magnitude (r):
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
Substituting [tex]\(x = 3\)[/tex] and [tex]\(y = -3\)[/tex]:
[tex]\[
r = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\][/tex]
2. Angle ([tex]\(\theta\)[/tex]):
[tex]\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\][/tex]
Substituting [tex]\(x = 3\)[/tex] and [tex]\(y = -3\)[/tex]:
[tex]\[
\theta = \tan^{-1}\left(\frac{-3}{3}\right) = \tan^{-1}(-1)
\][/tex]
The value of [tex]\(\tan^{-1}(-1)\)[/tex] is [tex]\(-45^\circ\)[/tex], but we must convert this to a positive angle within [tex]\(0^{\circ} < \theta < 360^{\circ}\)[/tex]. Since the point [tex]\((3, -3)\)[/tex] is in the fourth quadrant:
[tex]\[
\theta = 360^\circ - 45^\circ = 315^\circ
\][/tex]
### Step 2: Determine the Two Pairs
In polar coordinates, the point [tex]\((r, \theta)\)[/tex] can also be represented as [tex]\((-r, \theta + 180^\circ)\)[/tex], where [tex]\(\theta + 180^\circ\)[/tex] must also be within [tex]\(0^{\circ} < \theta < 360^{\circ}\)[/tex].
1. First Pair:
[tex]\[
(r, \theta) = \left(3\sqrt{2}, 315^\circ\right)
\][/tex]
2. Second Pair:
[tex]\[
(-r, \theta + 180^\circ)
\][/tex]
Since [tex]\(\theta = 315^\circ\)[/tex]:
[tex]\[
\theta + 180^\circ = 315^\circ + 180^\circ = 495^\circ
\][/tex]
We need to convert [tex]\(495^\circ\)[/tex] to a value within [tex]\(0^{\circ} < \theta < 360^{\circ}\)[/tex]:
[tex]\[
495^\circ - 360^\circ = 135^\circ
\][/tex]
So the second pair is:
[tex]\[
\left(-3\sqrt{2}, 135^\circ\right)
\][/tex]
### Step 3: Compare with Given Choices
Now, we compare our results with the given choices:
- a. [tex]\(\left(3 \sqrt{2}, 45^{\circ}\right),\left(-3 \sqrt{2}, 225^{\circ}\right)\)[/tex]
- b. [tex]\(\left(3 \sqrt{2}, 315^{\circ}\right),\left(-3 \sqrt{2}, 135^{\circ}\right)\)[/tex]
- c. [tex]\(\left(3 \sqrt{2}, 135^{\circ}\right),\left(-3 \sqrt{2}, 315^{\circ}\right)\)[/tex]
- d. [tex]\(\left(3 \sqrt{2}, 225^{\circ}\right),\left(-3 \sqrt{2}, 45^{\circ}\right)\)[/tex]
### Conclusion
The correct pair of polar coordinates for [tex]\((3, -3)\)[/tex] is:
- [tex]\(\left(3 \sqrt{2}, 315^{\circ}\right), \left(-3 \sqrt{2}, 135^{\circ}\right)\)[/tex]
Thus, the best answer is:
B.
