Question

Find the polar equation in rectangular form.

r=-16sin theta

Please no direct answers....I just need the formula(s) for doing this (:

Answer 1

Hint multiply both sides by r.

Answer 2

And then try to use some or all of the following:
\[r^2=x^2+y^2 \\ r \cos(\theta)=x \\ r \sin(\theta)=y \\ \tan(\theta)=\frac{y}{x}\]

Answer 3

@myininaya...I'm not sure what it is supposed to look like when multiplying r by (-16 sin theta)....

Answer 4

Do you have an example by chance doing this?

Answer 5

just r*(-16sin(theta))
or -16 r sin(theta) since multiplication is associative and commutative

and don't forgot you also need to multiply the other side by r

Answer 6

Okay,

So it would be

\[r^2=-16r~sin\theta\]

Answer 7

yes

Answer 8

now use the equations mentioned by me or the ones in the picture provided by @Dani_Rose
(they are the same )

Answer 9

Will I need to isolate the r then? rather than it being r^2?

Answer 10

you are trying to write in terms of x and y...
r^2 is...

Answer 11

r sin(theta) is ...

Answer 12

r sin theta would then be equal to the square root of (x^2+y^2), right?

Answer 13

wait...I think I'm going at that wrong.

Answer 14

did you read the equations I posted

Answer 15

oh...sorry

r sin theta is y.

Answer 16

these:
\[r^2=x^2+y^2 \\ r \cos(\theta)=x \\ r \sin(\theta)=y \\ \tan(\theta)=\frac{y}{x} \]


you are using those to write
\[r^2=-16 r \sin(\theta) \text{ in terms of } x \text{ and } y\]

Answer 17

you see that you can replace r^2 (the thing on the left hand side) with?
you see the r sin(theta) on the right hand side which can be replaced with?

Answer 18

and yes r sin(theta) is y ...so just replace that with y
and replace r^2 with...

Answer 19

okay, r^2 would be sqrt(x^2+y^2)

Answer 20

do you see the first equation I wrote?

Answer 21

r^2=x^2+y^2 ...

Answer 22

you shouldn't be involving a square root in that

Answer 23

So sorry, I don't know where my brain is today ;-;

Answer 24

so replace the r^2 and r sin(theta) with x^2+y^2 and y respectively and you are done writing as a Cartesian equation

Answer 25

so \[x^2+y^2=-16y\]?

Answer 26

yep

Answer 27

ah..

Answer 28

any questions?

Answer 29

Well, I was wondering how to get it into one of these forms...one second

Answer 30

this is a circle you can write in the form (x-h)^2+(y-k)^2=r^2

Answer 31

involves completing square

Answer 32

Ohhh, maybe that's what it is

Answer 33

\[x^2+y^2+16y=0\]
we don't have to worry with the x-stuff because the x part as already been written as a square
we need to do something about the y part of this...

Answer 34

OH!

Answer 35

I love this stuff...

Answer 36

so we just complete the square of \[y^2+16y+~~=0\]

Answer 37

what do we need to add to both sides to complete the square there

Answer 38

so (y+8)^2=64 right?

Answer 39

right so with out equation we need to retrice64 on both sides

Answer 40

Because you divide by 2 and then square that...add to both sides

Answer 41

\[x^2+y^2+16y=0 \\ x^2+y^2+16y+64=64 \\ x^2+(y+8)^2=64\]

Answer 42

Awesome, that makes a ton more sense. Thank you!!!

Answer 43

np