SOLUTION: find an equation in polar coordinates in 4x^2+y^2=4

Algebra ->  Coordinate-system -> SOLUTION: find an equation in polar coordinates in 4x^2+y^2=4      Log On


   



Question 1026746: find an equation in polar coordinates in 4x^2+y^2=4
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
When converting between rectangular and polar form,
either to or from, always draw this first:

 

Then write down the trig functions for the triangle:

sin%28theta%29%22%22=%22%22y%2Fr
cos%28theta%29%22%22=%22%22x%2Fr

Since we are converting from rectangular to polar, we solve
each of those for x and y.  

[If we were converting from polar to rectangular, we would
solve each for r.]

sin%28theta%29%22%22=%22%22y%2Fr
sin%28theta%29%2F1%22%22=%22%22y%2Fr
y%22%22=%22%22r%2Asin%28theta%29

cos%28theta%29%22%22=%22%22x%2Fr
cos%28theta%29%2F1%22%22=%22%22x%2Fr
x%22%22=%22%22r%2Acos%28theta%29

Substitute for x and y:

4x%5E2%2By%5E2%22%22=%22%224

4%28r%2Acos%28theta%29%29%5E2%2B%28r%2Asin%28theta%29%5E2%29%22%22=%22%224

Simplify and if reasonably easy, solve for r or rē:

4r%5E2%2Acos%5E2%28theta%29%2Br%5E2%2Asin%5E2%28theta%29%22%22=%22%224

r%5E2%284cos%5E2%28theta%29%2Bsin%5E2%28theta%29%5E%22%22%29%22%22=%22%224

r%5E2%22%22=%22%224%5E%22%22%2F%284cos%5E2%28theta%29%2Bsin%5E2%28theta%29%29

We can even make it simpler by substituting

1-cos%5E2%28theta%29 for sin%5E2%28theta%29

r%5E2%22%22=%22%224%5E%22%22%2F%284cos%5E2%28theta%29%2B1-cos%5E2%28theta%29%29

r%5E2%22%22=%22%224%5E%22%22%2F%283cos%5E2%28theta%29%2B1%29

Edwin