Let's solve the given problem step-by-step:
### (a) Plot the Point
Given point: [tex]\((4, 135^\circ)\)[/tex]
This point is located in the polar coordinate system where:
- [tex]\( r = 4 \)[/tex] is the radius, or the distance from the origin to the point.
- [tex]\( \theta = 135^\circ \)[/tex] is the angle measured counterclockwise from the positive x-axis to the point.
To plot the point:
1. Start from the origin.
2. Move along the positive x-axis and rotate counterclockwise by [tex]\(135^\circ\)[/tex].
3. From this angle, move outward a distance of 4 units.
### (b) Give Two Other Pairs of Polar Coordinates for the Point
The original point [tex]\((4, 135^\circ)\)[/tex] can be represented in multiple ways in polar coordinates. Here are two other representations:
1. First Representation:
- By adding [tex]\( 360^\circ \)[/tex] to the angle:
[tex]\((4, 135^\circ + 360^\circ) = (4, 495^\circ)\)[/tex]
2. Second Representation:
- Add [tex]\( 180^\circ\)[/tex] to the angle but reverse the sign of the radius:
The original point [tex]\((4, 135^\circ)\)[/tex] can also be represented as [tex]\((-4, 135^\circ + 180^\circ)\)[/tex]:
[tex]\((-4, 315^\circ)\)[/tex]
Therefore, the two other pairs of polar coordinates are [tex]\((4, 495^\circ)\)[/tex] and [tex]\((-4, 315^\circ)\)[/tex].
### (c) Convert to Rectangular Coordinates
To convert the polar coordinates [tex]\((4, 135^\circ)\)[/tex] to rectangular coordinates [tex]\((x, y)\)[/tex], we use the following formulas:
[tex]\[ x = r \cos(\theta) \][/tex]
[tex]\[ y = r \sin(\theta) \][/tex]
Here:
- [tex]\( r = 4 \)[/tex]
- [tex]\( \theta = 135^\circ \)[/tex]
We need to evaluate:
[tex]\[ x = 4 \cos(135^\circ) \][/tex]
[tex]\[ y = 4 \sin(135^\circ) \][/tex]
We know from trigonometric values:
- [tex]\(\cos(135^\circ) = -\frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\sin(135^\circ) = \frac{\sqrt{2}}{2}\)[/tex]
So,
[tex]\[ x = 4 \left(-\frac{\sqrt{2}}{2}\right) = -2\sqrt{2} \][/tex]
[tex]\[ y = 4 \left(\frac{\sqrt{2}}{2}\right) = 2\sqrt{2} \][/tex]
Thus, the rectangular coordinates are:
[tex]\[ (-2\sqrt{2}, 2\sqrt{2}) \][/tex]
### Summary of Answers
(a) The point [tex]\((4, 135^\circ)\)[/tex] can be plotted in the polar coordinate system.
(b) Two other pairs of polar coordinates representing the same point are:
- [tex]\((4, 495^\circ)\)[/tex]
- [tex]\((-4, 315^\circ)\)[/tex]
(c) The rectangular coordinates of [tex]\((4, 135^\circ)\)[/tex] are:
[tex]\[ (-2\sqrt{2}, 2\sqrt{2}) \][/tex]
So, based on options provided,
A. [tex]\((4,495^\circ)\)[/tex] is a correct other pair.
B. [tex]\((-4, 135^\circ)\)[/tex] is not a correct representation.
C. [tex]\((-4, 315^\circ)\)[/tex] is a correct other pair.
D. [tex]\((-4, 495^\circ)\)[/tex] is not a correct representation.
The rectangular coordinates of [tex]\((4, 135^\circ)\)[/tex] are:
[tex]\[ (-2\sqrt{2}, 2\sqrt{2}) \][/tex]
