Question

Polar Coordinates

Answer 1

@perl

Answer 2

we can use the conversion equations
$$ \Large{ r^2 = x^2 + y^2
\\ \theta = \arctan ( \frac y x )
\\x = r \cos \theta
\\~\\ y = r \sin \theta
}$$

Answer 3

okies

Answer 4

immediately we see the left side is r^2

Answer 5

hm so do we want to change it to rs?

Answer 6

$$
\Large {
x^2 + y^2 = ( x^2 + y^2 - x)^2
\\~\\
r^2 = ( r^2 - r \cos \theta ) ^2
}
$$

Answer 7

oo, I see! I hadn't thought of putting the x in

Answer 8

$$
\Large {
x^2 + y^2 = ( x^2 + y^2 - x)^2
\\~\\
r^2 = ( r^2 - r \cos \theta ) ^2
\\~\\
\sqrt{r^2} = \sqrt{( r^2 - r \cos \theta ) ^2 }
\\~\\
r = r^2 - r \cos \theta
\\~\\
r + r\cos \theta = r^2
\\~\\r ( 1 + \cos\theta ) = r^2
\\~\\ 1 + \cos\theta = r
}
$$

Answer 9

aaah

Answer 10

so that is the final polar coordinate?

Answer 11

well , i sort of ignored that there are two solutions to this

Answer 12

...er. it could be -r too yeah?

Answer 13

lets compare this to the rectangular graph , which we can graph on desmos

Answer 14

Alrighty.

Answer 15

https://www.desmos.com/calculator

Answer 16

he only graphs one limacon https://www.desmos.com/calculator/9thimuyhvf

Answer 17

ah ok, that makes it easy :)

Answer 18

so if I didn't have a graphing device I would substitute values for theta correct?

Answer 19

correct

Answer 20

Lovely, thanks!

Answer 21

$$\Large {
x^2 + y^2 = ( x^2 + y^2 - x)^2
\\~\\
r^2 = ( r^2 - r \cos \theta ) ^2
\\~\\
\sqrt{r^2} = \sqrt{( r^2 - r \cos \theta ) ^2 }
\\~\\
\pm r = \pm( r^2 - r \cos \theta )
\\~\\
r = r^2 - r \cos \theta
\\~\\
r + r\cos \theta =~ r^2
\\~\\r \cdot ( 1 + \cos\theta ) =~ r^2
\\~\\ (1 + \cos\theta) = ~ r
}$$

Answer 22

they would cancel out?

Answer 23

yes

Answer 24

hrm

Answer 25

if you want to be really fastidious , you can use absolute value

Answer 26

$$\Large {
x^2 + y^2 = ( x^2 + y^2 - x)^2
\\~\\
r^2 = ( r^2 - r \cos \theta ) ^2
\\~\\
\sqrt{r^2} = \sqrt{( r^2 - r \cos \theta ) ^2 }
\\~\\
| ~r~ |= |~ r^2 - r \cos \theta~ |
\\~\\
r = r^2 - r \cos \theta
\\~\\
r + r\cos \theta =~ r^2
\\~\\r \cdot ( 1 + \cos\theta ) =~ r^2
\\~\\ (1 + \cos\theta) = ~ r
}$$

but i don't see what you would gain from doing that

Answer 27

lol xD I think that's just going above and beyond now :P

Answer 28

for experts only!

Answer 29

thats really cool, hanging plants

Answer 30

in your pic