How do you convert the Cartesian coordinates (0, -5) to polar coordinates?

Question

Serenity Jones · Accepted Answer

#5 angle - 90^@# Cartesian co-ordinates (x;y) ma be converted to polar co-ordinates #(r;theta)# as follows : #r=x^2+y^2# #theta=tan^(-1)(y/x)# Plotting the point and then joining a line from the origin to the point is also useful since this line will have length r, and the angle the line makes with the x-axis will be #theta# So in this case, r = 5 and #theta=-90^@#

Elijah Allen · Answer

undefined To convert Cartesian coordinates to polar coordinates, follow these steps:

1. Identify the given point's $x$ and $y$ coordinates.
2. Use the formulas: $r = \sqrt{x^2 + y^2}$ for the distance from the origin and $θ = \arctan{\frac{y}{x}}$ for the angle measured counterclockwise from the positive x-axis to the point.

For the point (0, -5):

1. $x = 0$ and $y = -5$.
2. Calculate $r$ using $r = \sqrt{0^2 + (-5)^2} = \sqrt{0 + 25} = \sqrt{25} = 5$.
3. Calculate $θ$ using $θ = \arctan{\frac{-5}{0}}$.
   Since $x = 0$ (which corresponds to the y-axis), the angle $θ$ is either $π/2$ (if $y < 0$) or $-π/2$ (if $y > 0$).
   In this case, since $y = -5$, $θ = \arctan{\frac{-5}{0}} = -\frac{π}{2}$.

So, the polar coordinates are $(5, -\frac{π}{2})$.