Answer to Question #129236 in Analytic Geometry for Sym

Question

Express
(2,π/6 in rectangular coordinates

Express r=1+2cos∅ from polar to cartesian form

Answer 1

I)Express $(2,\frac{\pi}{6})$ in rectangular coordinates:

$x=rcos\theta,$

$y=rsin\theta;$

$r=2,\space \theta=\frac{\pi}{6}$

$x=2*cos(\frac{\pi}{6}),$

$y=2*sin(\frac{\pi}{6}).$

$x=2*\frac{\sqrt{3}}{2}=\sqrt{3},$

$y=2*\frac{1}{2}=1.$

Answer:$x=\sqrt{3},\space y=1.$

II)Express $r=1+2cos\theta$ from polar to cartesian form.

$x=rcos\theta,$

$y=sin\theta.$

$r=\sqrt{x^2+y^2}$

$r=1+2cos\theta$

$2cos\theta=r-1$

$2(\frac{x}{r})=r-1$

$2x=r^2-r$

$2x=(x^2+y^2)-\sqrt{x^2+y^2}$

Answer:

$x^2+y^2=2x+\sqrt{x^2+y^2}.$