Problem 18 - Essential Calculus

Question

Use the angle feature of a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. $$ (0,-5) $$

Step-by-Step Solution

Answer

The point (0, -5) in rectangular coordinates corresponds to (-5, -π/2 rad) in polar coordinates.

Step 1: Determine the Radial Coordinate

For any point (x,y), the radial coordinate r can be found using the Pythagorean theorem: r = sqrt(x^2 + y^2). In this case, r = sqrt(0^2 + (-5)^2) = sqrt(25) = 5. Because the point is below the x-axis, the radial coordinate will be -5 (distance is 5 units from origin but directed along the negative y-axis).

Step 2: Calculate the Angular Coordinate

θ = atan2(y, x), where atan2 is the two-argument arctangent function. But since x = 0, directly plugging into the formula would give a division by zero error. But as a special case, if x=0 and y is negative, then the angle θ is -90 degrees. Convert this to radians by multiplying by π/180, θ = (-90 * π) / 180 = -π/2 rad.