Final answer:
The polar coordinates of the rectangular coordinates (2, -23) are (23.086, 4.8775), where we calculated the radial distance r using the Pythagorean theorem and found the angle θ by adding 2π to the arctan of y/x to get a value in the range of 0 to 2π.
Step-by-step explanation:
To find the polar coordinates of the rectangular coordinates (2, -23), we first calculate the radial distance r and then determine the angle θ. The radial distance r is the distance from the origin to the point and is calculated using the Pythagorean theorem:
r = √(x2 + y2)
For the coordinates (2, -23), we have:
r = √(22 + (-23)2)
= √(4 + 529)
= √533
= 23.086
The angle θ is the angle measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point. Since the point lies in the fourth quadrant (where x is positive, and y is negative), we use the arctangent to find θ:
θ = arctan(y/x)
But we need to add 2π since the angle is in the fourth quadrant and we want 0 ≤ θ < 2π.
θ = arctan(-23/2) + 2π
= arctan(-11.5) + 2π
= -1.40564764938027 + 2π
= 4.87753942386509
So the polar coordinates are (23.086, 4.8775), where r is positive, and 0 ≤ θ < 2π.