convert polar equation to rectangular coordinates:
r^2=4sin2(theta)
conversion formulas
r^2 = x^2 + y^2
sinØ = y/r
cosØ = x/r
r^2 = 4 sin 2Ø
r^2 = 4(2sinØcosØ) = 8sinØcosØ
x^2 + y^2 = 8(y/r)(x/r)
= 8xy/r^2
x^2 + y^2 = 8xy/(x^2 + y^2)
(x^2 + y^2)^2 = 8xy
To convert the polar equation r^2 = 4sin^2(theta) to rectangular coordinates, we can use the following relationships:
x = r * cos(theta)
y = r * sin(theta)
Let's substitute these values into the equation to convert it into rectangular form.
First, rewrite the polar equation as r^2 = 4 * sin^2(theta):
r^2 = 4sin^2(theta)
Now, substitute the values of x and y:
x^2 + y^2 = 4y^2
Next, simplify the equation:
x^2 + y^2 = 4y^2
Rearrange the equation:
x^2 + y^2 - 4y^2 = 0
Combine like terms:
x^2 + (1-4)y^2 = 0
Simplify:
x^2 - 3y^2 = 0
Hence, the polar equation r^2 = 4sin^2(theta) can be represented in rectangular coordinates as x^2 - 3y^2 = 0.