SOLUTION: Convert the given polar equation to a Cartesian equation. (Use the following as necessary: x and y.) r = 4 sin(𝜃)

Algebra ->  Trigonometry-basics -> SOLUTION: Convert the given polar equation to a Cartesian equation. (Use the following as necessary: x and y.) r = 4 sin(𝜃)      Log On


   

Question 1204941: Convert the given polar equation to a Cartesian equation. (Use the following as necessary: x and y.)
r = 4 sin(𝜃)
Found 4 solutions by ikleyn, MathLover1, greenestamps, math_tutor2020:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Convert the given polar equation to a Cartesian equation. (Use the following as necessary: x and y.)
r = 4 sin(𝜃)
~~~~~~~~~~~~~~~~~~~~~~~~~ 

The Cartesian equation is

    sqrt%28x%5E2%2By%5E2%29 = 4%2A%28y%2Fr%29     

or

    sqrt%28x%5E2%2By%5E2%29 = %284y%29%2Fsqrt%28x%5E2%2By%5E2%29%29 


It is a Cartesian equation as a primary translation.   



sqrt%28x%5E2%2By%5E2%29  represents "r";  right side represents  4%2Asin%28theta%29.


You may further multiply both sides by sqrt%28x%5E2%2By%5E2%29  to get

    x%5E2%2By%5E2 = 4y.


It is a Cartesian equation in its simplified form.

Solved.



Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

r+=+4sin%28theta%29
we know that:
x=r%2Acos%28theta%29+ => cos%28theta%29=x%2Fr.....eq.1
y=r%2Asin%28theta%29+=>sin%28theta%29=y%2Fr....eq.2
tan%5E-1%28theta%29=y%2Fx....eq.3
r=sqrt%28x%5E2%2By%5E2%29 => r%5E2=x%5E2%2By%5E2....eq.4


then, substituting sin%28theta%29 from eq.2 given equation becomes
r+=+4%28y%2Fr%29......both sides multiply by r
r%5E2+=+4y ....substitute r%5E2 from eq.4
x%5E2%2By%5E2+=+4y+
x%5E2%2By%5E2+-4y+=0 => a Cartesian equation

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


r=4sin%28theta%29

Knowing that y=r%2Asin%28theta%29, multiply both sides of the equation by r:

r%5E2=4rsin%28theta%29

Then use r%5E2=x%5E2%2By%5E2%29 and rsin%28theta%29=y to convert the equation to a Cartesian equation:

x%5E2%2By%5E2=4y

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

The other tutors have great answers. I'll extend their scratch work to complete the square for the y term, and use it to determine the center and radius of this circle.

In the fourth step shown below, I add and subtract 4.
The 4 comes from taking half of the y coefficient, and then squaring it.

x^2 + y^2 = 4y
x^2 + y^2 - 4y = 0
x^2 + (y^2 - 4y) = 0
x^2 + (y^2 - 4y + 4 - 4) = 0
x^2 + (y^2 - 4y + 4) - 4 = 0
x^2 + (y-2)^2 - 4 = 0
x^2 + (y-2)^2 = 4

Compare that to the circle template (x-h)^2 + (y-k)^2 = r^2
We determine that h = 0, k = 2, r = 2
This circle has its center at (h,k) = (0,2) and has radius r = 2.
Desmos can be used to confirm the answer
https://www.desmos.com/calculator/os2lf1m92p
Click the wrench icon in the upper right corner to go from cartesian mode to polar mode.

Even though it appears your teacher may not be asking for the circle's center and radius, it's still good practice to be able to find it.