- anonymous

1. Algebraically transform r = -8sinθ into a Cartesian equation of a circle (in standard form).

- jamiebookeater

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- amistre64

algebraically eh.....

- amistre64

r = -8sin(t) is a straight line right?

- anonymous

no, it's a circle

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## More answers

- amistre64

I see it :)...polars still get me a little lol

- anonymous

Ha same here

- amistre64

lets do I table to see where it going ...can we do that?
t = 0,45,90,360....

- amistre64

when its at 90; or straight up, we get r = -8 so this is a circle about the y axis with a radius of 8 centerd 4 below the origin...

- amistre64

but I am prolly messing that up still....

- amistre64

nah..im right lol

- amistre64

##### 1 Attachment

- anonymous

well in my textbook theres a question similar to this, but i dont know how to get the last equation.
r=6sinθ
r²=6rsinθ
x²y²=6y
x²-6y+9+x²=9
(y-3)²+x²=3²

- amistre64

yeah, I was gettin to that :) just wanted to make sure I had an answer key to go by ;)

- anonymous

OOHHH sorry

- amistre64

our equation is going to end up as:
x^2 + (y+4)^2 = 16
r = -8sin(t)

- amistre64

r = -8sin(t) ; muliply by r
r^2 = -8rsin(t) ; convert to cartesians
x^2 + y^2 = -8y ; +8y
x^2 + y^2 +8 = 0 ; complete square for "y"
x^2 + y^2 +8y +16 = 16 ; convert to circle equation
x^2 + (y+4)^2 = 16 ; tada!!!

- amistre64

you see thats a typo ...8y*

- anonymous

At which step? and where did the 16 come from?

- amistre64

by now you should be familiar with a process called "completing the square". remember it?

- amistre64

step 4 I put in 8 instead of 8y ;)

- anonymous

Oh ok I see the typo

- amistre64

a complete square can be transformed back and forth between to ...... forms.

- amistre64

(x+4)^2 = x^2 +8x +16 right?
we can easily move back and forth between these equations since the are equal to each other...

- amistre64

but what is usually missing in order to "complete" a sqaure is this:
(x+___)^2 = x^2 +8x + ______

- amistre64

we need 2 numbers that are exactly the same, that add to get 8. what numbers are they?

- anonymous

4

- amistre64

good :) 4+4 = 8, that will satisfy that middle term; now we multiply 4*4 to get the last term.
4^2 = 16
(x+4)^2 = x^2 +8x + 16 you see where we got it now?

- anonymous

yeah

- amistre64

x^2 + 6x + ____
what would we complete the square with here?

- anonymous

(x+3)²=x²+6x+12?

- amistre64

real close....look again. 3+3 = 6 3*3 != 12

- amistre64

3*3 = ?

- anonymous

9

- amistre64

good :) we need to add 9 to "complete" the square
what happens when you add a number to one side of an equation? what do we do to the other side?

- anonymous

Add it on the other side

- amistre64

Exactly :) Do you see where I did that in the problem?
x^2 + y^2 +8 = 0 ; complete square for "y"
x^2 + y^2 +8y +16 = 16

- amistre64

i really should fix that typo lol

- anonymous

Haha no. That's where I'm stuck.

- amistre64

x^2 + [ y^2 +8y +____] = 0 ; complete square for "y"
what number do we use to "complete" the square for y?

- anonymous

16

- amistre64

its always gonna be half the middle term and then square it

- amistre64

8/2 = 4 -> 4^2 = 16 :) good

- amistre64

x^2 + [y^2 +8y +16] = 0 + 16
^^^ ^^^
we add 16 to both sides righ there right?

- anonymous

Yeah

- amistre64

then we just clean it up:
x^2 + (y+4)^2 = 16

- amistre64

which agrees with what we drew in the first place ;)

- anonymous

Where did the 8y go then?

- amistre64

remember we can move between the forms of a complete square?
(y+4)^2 = y^2 +8y +16
we just use the one for the other.... they are identical in value, they only look different in form.

- anonymous

Ohhh ok. I get it now!!

- amistre64

once we got a "complete" square, we use it to clean up the equation ;)

- anonymous

Ok, I finally understand this now! Thanks for helping me out!

- amistre64

youre welcome :)

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