Question

In Exercises \(5-8,\) convert the point from rectangular coordinates to cylindrical coordinates. (0,5,1)

Short answer

Therefore, the cylindrical coordinates of the point (0,5,1) are (5, \(\pi/2\), 1).

Step-by-step solution

Calculate the radius r

Firstly, calculate \(r\), the distance from the point to the z-axis using the formula \(r = \sqrt{x^2 + y^2}\). Substituting x = 0 and y = 5, we get \(r = \sqrt{0^2 + 5^2} = 5\).

Calculate the angle θ

Then we determine the angle θ between the positive x-axis and the line connecting the point to the origin in the xy-plane using the formula \(\theta = arctan(y / x)\). But, as x = 0 in this case, recall the special case: if x = 0, then either \(\theta = 90°\) when y > 0 or \(\theta = 270°\) when y < 0. In this case y > 0, so \(\theta = 90°\), converting degrees into radians, gives \(\theta = \frac{\pi}{2}\).

Retain z

Finally, copy the z-coordinate unchanged from the rectangular point, so z = 1.