How would one convert the cartesian expression y=1/x to polar form?

Question

How would one convert the cartesian expression y=1/x to polar form? I'd really appreciate a step-by-step solution so I can apply the same principle to other problems. Thanks!

Answer 1

By definition x=rcosθ ,y=rsinθ. where r is the radial distance from the origin, and theta is the counterclockwise angle from the x-axis. In terms of x and y $$r=\sqrt{x^2+y^2}$$ and $$θ=tan^{-1}{(y/x)}$$ Can you do the substitution?

Answer 2

Just substitute $x=r \cos \theta$ and $y=r \sin \theta$: $$ y=1/x \rightarrow r \sin \theta=\frac{1}{r \cos \theta}\rightarrow r^2=\frac{1}{ \sin \theta \cos \theta}=\frac{2}{ \sin 2\theta} $$ so: $$ r=\pm\sqrt{\frac{2}{ \sin 2\theta}} $$