# Finding center of circle with Polar Coordinates

## Homework Statement

r=7sin(∅)
find the center of the circle in Cartesian coordinates and the radius of the circle

## The Attempt at a Solution

My math teacher is impossible to understand >.< and then the stupid homework is online and crap blah this class but I REALLY want to understand the material well.. anyway done venting to earn your pity but I REALLY don't know how to do this one and I locked my textbook in my car lol but since I know that isn't acceptable on physics forums I will try >.<

Okay I really don't know.. maybe I shouldn't have started this thread yet... I'll go research online and edit this post soon if anyone wants to be cool and post hints even though i'm breaking the rules that'd be cool.

my attempt:
r=7sin(∅)
x=rcos(∅)
y=rsin(∅)

x^2 + y^2 = r^2

plugging r in to the x and y equations and squaring...
49(sin^2(∅)*cos^2(∅)) + 49sin^4(∅) = 49sin^2(∅)
(sin^2(∅)*cos^2(∅)) + sin^4(∅) = sin^2(∅)
cos^2(∅) + sin^2 (∅) = 1
is what that reduces to.... alright sooo that's kind of cool I accidently did that but I got no closer to finding the center.. lol ;-/

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CAF123
Gold Member
You know that x2+ y2=r2, so you can find what r is. There is also a very simple equation for sin∅ from elementary trig which you can use.

You know that x2+ y2=r2, so you can find what r is. There is also a very simple equation for sin∅ from elementary trig which you can use.
Should I be getting the answer that r = 1?
If so any tips on how I go about finding the center?

CAF123
Gold Member
Should I be getting the answer that r = 1?
If so any tips on how I go about finding the center?
How did you get r=1?

Make a right angle triangle with base of length x and height y. Let ∅ be the angle between hypotenuse and base. Find sin∅ in terms of x and y. You should also be able to find r, so sub all these expressions into r = 7sin∅.

eumyang
Homework Helper
OP: I mentioned this before, but please stop using the empty set symbol (∅) for theta (θ). Theta can be found in the 3rd row of the "Quick Symbols" box. Sorry to nitpick.

I would like to propose a slightly different way to solve the problem. You've given us the conversion equations:
x = r cos θ
y = r sin θ
x2 + y2 = r2

Take the original equation
r = 7 sin θ
and multiply both sides by r. Start substituting. You'll need to complete the square at some point.

How did you get r=1?

Make a right angle triangle with base of length x and height y. Let ∅ be the angle between hypotenuse and base. Find sin∅ in terms of x and y. You should also be able to find r, so sub all these expressions into r = 7sin∅.
sin(θ)= y/r

x=rcos(θ)
x/cos(θ)=r

sin(θ) = y/(x/cos(θ))
sin(θ) = ycos(θ)/x

r=7sin(θ)
r= 7ycos(θ))/x

ahh.. am I on the right track here?
maybe if I backtrack to
sin(θ) = ycos(θ)/x
tan(θ) = y/x

OP: I mentioned this before, but please stop using the empty set symbol (∅) for theta (θ). Theta can be found in the 3rd row of the "Quick Symbols" box. Sorry to nitpick.

I would like to propose a slightly different way to solve the problem. You've given us the conversion equations:
x = r cos θ
y = r sin θ
x2 + y2 = r2

Take the original equation
r = 7 sin θ
and multiply both sides by r. Start substituting. You'll need to complete the square at some point.
r^2 = r7sinθ

r=x/cos(θ)
r=y/sin(θ)
x^2/cos^2 = y/sin(θ) * 7sinθ
x^2/cos^2 = 7y

idk what i'm doing

Mark44
Mentor
r^2 = r7sinθ

r=x/cos(θ)
r=y/sin(θ)
x^2/cos^2 = y/sin(θ) * 7sinθ
x^2/cos^2 = 7y

idk what i'm doing
Starting from r2 = 7rsin(θ), replace r and θ to get the equation in Cartesian form.

You know (I hope!) that:
r2 = x2 + y2
r sin(θ) = y
r cos(θ) = x

• 1 person
eumyang
Homework Helper
r^2 = r7sinθ

r=x/cos(θ)
r=y/sin(θ)
x^2/cos^2 = y/sin(θ) * 7sinθ
x^2/cos^2 = 7y

idk what i'm doing
I see. I don't want to sound mean, but you need to do some review. And you're thinking too hard. From
r2 = 7r sin θ
all you need to do is replace the left side with x2 + y2 and replace the "r sin θ" on the right side with y, and you get
x2 + y2 = 7y.
Do you know the equation for a circle with radius r and center at (h, k)?

• 1 person
CAF123
Gold Member
sin(θ)= y/r
Yes, but you can also express r in terms of x and y. r2= x2+ y2 so find r.

x=rcos(θ)
x/cos(θ)=r
What you want to achieve is the polar expression written completely in Cartesian. Writing r=x/cosθ is fine, but this does not eliminate θ, so it is not helpful here.

Your final result should have just x and y.

I see. I don't want to sound mean, but you need to do some review. And you're thinking too hard. From
r2 = 7r sin θ
all you need to do is replace the left side with x2 + y2 and replace the "r sin θ" on the right side with y, and you get
x2 + y2 = 7y.
Do you know the equation for a circle with radius r and center at (h, k)?
(x-h)^2 + (y-k)^2 = r^2 ???
I'm having problems solving the formula to make it fit in this form,, too many y's -_-

ehild
Homework Helper
(x-h)^2 + (y-k)^2 = r^2 ???
I'm having problems solving the formula to make it fit in this form,, too many y's -_-
Do not mix r, the polar coordinate, with the radius of the circle.

In Cartesian coordinates, the equation of a circle is ˙(x-h)2+(y-k)2=R2.

You already got the equation of the circle in the form x2 + y2 = 7y which is equivalent with x2 -7y+y2 = 0. Use the method completing the square.

ehild

Do not mix r, the polar coordinate, with the radius of the circle.

In Cartesian coordinates, the equation of a circle is ˙(x-h)2+(y-k)2=R2.

You already got the equation of the circle in the form x2 + y2 = 7y which is equivalent with x2 -7y+y2 = 0. Use the method completing the square.

ehild
the x coordinate of the circle is centered at zero?
but how do I complete the square of y^2-7y? y(y-7)? that's not the correct format of (7-k)^2

also you say don't mix the polar coordinate r with the circle radius R, but to get the equation
x^2+y^2 = 7y
I solved for r in
r=7sin(theta)
r^2=r7sin(theta)
r^2=7y (y = rsin(theta)

so I plugged that r into
x^2 + y^2 = R^2, that's not correct?

CAF123
Gold Member
the x coordinate of the circle is centered at zero?
Yes
but how do I complete the square of y^2-7y? y(y-7)? that's not the correct format of (7-k)^2
Write ##y^2 - 7y = (y-b)^2 + c##. Expand the RHS and equate coefficients to solve for b and c.

• 1 person
Yes

Write ##y^2 - 7y = (y-b)^2 + c##. Expand the RHS and equate coefficients to solve for b and c.
-5y = b^2 + c. I don't understand what c is here. I assume b is the y coordinate of the center of the circle. How do I solve for two variables using one equation?

ehild
Homework Helper
the x coordinate of the circle is centered at zero?
but how do I complete the square of y^2-7y? y(y-7)? that's not the correct format of (7-k)^2

also you say don't mix the polar coordinate r with the circle radius R, but to get the equation
x^2+y^2 = 7y
I solved for r in
r=7sin(theta)
r^2=r7sin(theta)
r^2=7y (y = rsin(theta)

so I plugged that r into
x^2 + y^2 = R^2, that's not correct?
No, r is not the same as R. Forget about the polar coordinates. You have the equation for a curve in Cartesian coordinates x,y:

x2+y2-7y=0

How do you know that it is the equation of a circle?

Try to plot it. If y=0 what is x? If x=0, what is y?

ehild

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• 1 person
No, r is not the same as R. Forget about the polar coordinates. You have the equation for a curve in Cartesian coordinates x,y:

x2+y2-7y=0

How do you know that it is the equation of a circle?

Try to plot it. If y=0 what is x? If x=0, what is y?

ehild
But to get that I plugged the r from r=7sinθ into it... how did I use that r to get the equation if it's not the same R as x^2 + y^2 = R^2??

Cuz I multipled both sides for r and then plugged in y..
r^2 = 7rsinθ
r^2 = 7y.
x^2 + y^2 = R^2
x^2 + y^2 = 7y
you are okay with doing that to get the equation even though they aren't the same R? i'm so confused

Oh my god you are my savior for posting that link. Thank you everyone in this thread I wouldn't be passing calculus without you.