Question

How do you convert `(3, - 4)` into polar coordinates?

Answer 1

`(r,theta)=(5,2pi+arctan(-3/4))` (approximately `=(5,5.64)`

Explanation:

The radius `r` is given by the Pythagorean Theorem as
`color(white)("XXX")r=sqrt((-4)^2+3^2) = 5`

The value of `theta` is based on the `arctan(y/x)` but is dependent upon the quadrant in which `(x,y)` is positioned. This is because the `arctan` function returns a value in the range `[-pi/2,+pi/2]` relative to the X-axis.

#{: (color(black)("Quadrant"),,color(white)("XXX")color(black)(theta)), (bar(color(white)("XXXXXX")),,bar(color(white)("XXXXXX"))), (I,,arctan(y/x)), (II,,arctan(y/x)+pi), (III,,arctan(y/x)+pi), (IV,,arctan(y/x)+2pi) :}#