how would i convert r = 1 + 2 cos(2θ) into the rectangular coordinates?

Question

ganeshie8 · Answer

$\large  r = 1+ 2\cos (2\theta) = 1 +  2[2\cos^2\theta - 1] = 4\cos^2\theta  - 1$

$\large \implies r^3 =   4r^2\cos^2\theta - r^2 $

ganeshie8 · Answer

plugin 
$x = r\cos \theta $
$r^2 = x^2+y^2$

anonymous · Answer

does that mean r^3=4x^2 -x^2 +y^2?

ganeshie8 · Answer

Yep ! replace the $r^3$ on left hand side also

anonymous · Answer

how would i do that?

ganeshie8 · Answer

$\large  r^3 =   3x^2  - y^2 $

ganeshie8 · Answer

square both sides

ganeshie8 · Answer

$\large  (r^3)^2 =  ( 3x^2  - y^2)^2 $

ganeshie8 · Answer

$\large  (r^2)^3 =  ( 3x^2  - y^2)^2 $

ganeshie8 · Answer

$\large  (x^2+y^2)^3 =  ( 3x^2  - y^2)^2 $

ganeshie8 · Answer

we're done.

anonymous · Answer

but how would i do the graph for that equation?

ganeshie8 · Answer

ganeshie8 · Answer

$r = 1 + 2 \cos(2\theta )$

when $\theta =  0^{\circ}$, $r =  1+2(1) = 3$
so the curve starts on right side at r = 3

ganeshie8 · Answer

may be, plot few more points till u get some feel of how it is looping

anonymous · Answer

oh right! thanks

ganeshie8 · Answer

goodluck ! @BSwan  is expert in sketching polar curves, Bswan give her a hand if you have time :)

anonymous · Answer

she  wanna  draw it ?

ganeshie8 · Answer

ask her

ganeshie8 · Answer

@swift_13

anonymous · Answer

oh its ohk i found out how to do it,thanks though

anonymous · Answer

:D
good luck !

anonymous · Answer

even thought i love this curve xD