Answer:
To find the rectangular coordinates of the polar coordinates (3, -2), we can use the following formulas:
x = r cos(theta)
y = r sin(theta)
where r is the distance from the origin to the point, and theta is the angle between the positive x-axis and the line connecting the origin and the point, measured in radians.
In this case, r = 3 and theta = -2. To convert the angle to radians, we can add 2 pi to it, since negative angles represent angles measured clockwise from the positive x-axis in polar coordinates. So, theta = -2 + 2 pi = 4.283 radians.
Using the formulas, we get:
x = 3 cos(4.283) ≈ -1.008
y = 3 sin(4.283) ≈ -2.573
Therefore, the rectangular coordinates of the polar coordinates (3, -2) are approximately (-1.008, -2.573).
To find the polar coordinates of the rectangular coordinates (2, 5pi/4), we can use the following formulas:
r = sqrt(x^2 + y^2)
theta = atan2(y, x)
where x and y are the rectangular coordinates of the point.
In this case, x = 2 and y = 0. To find the angle, we can use the fact that the point is in the fourth quadrant, where x is positive and y is negative. Therefore, the angle is pi + 5pi/4 = 9pi/4 radians.
Using the formulas, we get:
r = sqrt(2^2 + 0^2) = 2
theta = atan2(0, 2) = 0
Therefore, the polar coordinates of the rectangular coordinates (2, 5pi/4) are (2, 9pi/4).