anonymous
  • anonymous
1. Algebraically transform r = -8sinθ into a Cartesian equation of a circle (in standard form).
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
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amistre64
  • amistre64
algebraically eh.....
amistre64
  • amistre64
r = -8sin(t) is a straight line right?
anonymous
  • anonymous
no, it's a circle

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amistre64
  • amistre64
I see it :)...polars still get me a little lol
anonymous
  • anonymous
Ha same here
amistre64
  • amistre64
lets do I table to see where it going ...can we do that? t = 0,45,90,360....
amistre64
  • 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
  • amistre64
but I am prolly messing that up still....
amistre64
  • amistre64
nah..im right lol
amistre64
  • amistre64
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anonymous
  • 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
  • amistre64
yeah, I was gettin to that :) just wanted to make sure I had an answer key to go by ;)
anonymous
  • anonymous
OOHHH sorry
amistre64
  • amistre64
our equation is going to end up as: x^2 + (y+4)^2 = 16 r = -8sin(t)
amistre64
  • 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
  • amistre64
you see thats a typo ...8y*
anonymous
  • anonymous
At which step? and where did the 16 come from?
amistre64
  • amistre64
by now you should be familiar with a process called "completing the square". remember it?
amistre64
  • amistre64
step 4 I put in 8 instead of 8y ;)
anonymous
  • anonymous
Oh ok I see the typo
amistre64
  • amistre64
a complete square can be transformed back and forth between to ...... forms.
amistre64
  • 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
  • amistre64
but what is usually missing in order to "complete" a sqaure is this: (x+___)^2 = x^2 +8x + ______
amistre64
  • amistre64
we need 2 numbers that are exactly the same, that add to get 8. what numbers are they?
anonymous
  • anonymous
4
amistre64
  • 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
  • anonymous
yeah
amistre64
  • amistre64
x^2 + 6x + ____ what would we complete the square with here?
anonymous
  • anonymous
(x+3)²=x²+6x+12?
amistre64
  • amistre64
real close....look again. 3+3 = 6 3*3 != 12
amistre64
  • amistre64
3*3 = ?
anonymous
  • anonymous
9
amistre64
  • 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
  • anonymous
Add it on the other side
amistre64
  • 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
  • amistre64
i really should fix that typo lol
anonymous
  • anonymous
Haha no. That's where I'm stuck.
amistre64
  • amistre64
x^2 + [ y^2 +8y +____] = 0 ; complete square for "y" what number do we use to "complete" the square for y?
anonymous
  • anonymous
16
amistre64
  • amistre64
its always gonna be half the middle term and then square it
amistre64
  • amistre64
8/2 = 4 -> 4^2 = 16 :) good
amistre64
  • amistre64
x^2 + [y^2 +8y +16] = 0 + 16 ^^^ ^^^ we add 16 to both sides righ there right?
anonymous
  • anonymous
Yeah
amistre64
  • amistre64
then we just clean it up: x^2 + (y+4)^2 = 16
amistre64
  • amistre64
which agrees with what we drew in the first place ;)
anonymous
  • anonymous
Where did the 8y go then?
amistre64
  • 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
  • anonymous
Ohhh ok. I get it now!!
amistre64
  • amistre64
once we got a "complete" square, we use it to clean up the equation ;)
anonymous
  • anonymous
Ok, I finally understand this now! Thanks for helping me out!
amistre64
  • amistre64
youre welcome :)

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