# Show that r^2 = x^2 + y^2

## Homework Statement

Consider z=f(x,y), where x=rcosθ and y=rsinθ
(This is a multi part multi variable calculus assignment question). I've just derived dz/dr and now I'm asked... to show that $r^{2}=x^{2}+y^{2}, \theta=a\tan(\frac{y}{x})$

## The Attempt at a Solution

I've drawn up a triangle with r as the hypotenuse, x on the x axis, y on the y axis.

and then x=rcosθ=r * x/r = x by trig laws and then $r^{2}=x^{2}+y^{2}$ by pythagoras theorem etc

this seems too easy. is there some other way to show this?

SteamKing
Staff Emeritus
Homework Helper
Not unless you want to prove the Pythagorean Theorem itself.

LCKurtz
Homework Helper
Gold Member

## Homework Statement

Consider z=f(x,y), where x=rcosθ and y=rsinθ
(This is a multi part multi variable calculus assignment question). I've just derived dz/dr and now I'm asked... to show that $r^{2}=x^{2}+y^{2}, \theta=a\tan(\frac{y}{x})$

That doesn't have anything to do with ##z = f(x,y)##. It's just standard polar coordinate equations.$$x^2 + y^2 = r^2\cos^2\theta + r^2\sin^2\theta = r^2$$ $$\frac y x = \frac{r\sin\theta}{r\cos\theta}=\tan\theta$$

yeah that's why I was confused. I don't get how I'm showing that by just plugging it into equations and referring to the polar equations.