Topic: $Calculus \ 2$, polar conversions

Q: Identify name & Cartesian equation for: r = 2$\tan \theta \sec \theta$

Any graphing calculator can quickly show this is a parabola the one I've sketched below, but I'm trying to understand the process here. These are the facts I know of:

$r=\sqrt{x^2+y^2}$
$\theta = \tan^{-1}(\frac{y}{x})$
x = r cos θ
y = r sin θ

What's the trick here? Can somebody show me? (I'd be more than happy to give out a medal if you can do so) There are a few other problems related to this one I'm working on so a technique to be learned is the goa

Question

Topic:  $Calculus \ 2$, polar conversions

Q:  Identify name & Cartesian equation for: r = 2$\tan \theta \sec \theta$ 

Any graphing calculator can quickly show this is a parabola the one I've sketched below, but I'm trying to understand the process here.  These are the facts I know of:

$r=\sqrt{x^2+y^2}$
$\theta = \tan^{-1}(\frac{y}{x})$
x = r cos θ
y = r sin θ

What's the trick here?  Can somebody show me?  (I'd be more than happy to give out a medal if you can do so) There are a few other problems related to this one I'm working on so a technique to be learned is the goa

anonymous · Answer

$$\large \sqrt{x^2+y^2}=\tan(\tan^{-1}\frac{y}{x})\sec(\tan^{-1}\frac{y}{x})$$

But then um... lol what?

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