Certainly! Let's convert the equation [tex]\(x^2 = 4y\)[/tex] from rectangular coordinates [tex]\((x, y)\)[/tex] to polar coordinates [tex]\((r, \theta)\)[/tex].
1. Start with the original equation:
[tex]\[
x^2 = 4y
\][/tex]
2. Recall the relationships between rectangular and polar coordinates:
[tex]\[
x = r \cos(\theta)
\][/tex]
[tex]\[
y = r \sin(\theta)
\][/tex]
3. Substitute [tex]\(x\)[/tex] and [tex]\(y\)[/tex] with their polar coordinate expressions:
[tex]\[
(r \cos(\theta))^2 = 4 (r \sin(\theta))
\][/tex]
4. Simplify the left side of the equation:
[tex]\[
r^2 \cos^2(\theta) = 4r \sin(\theta)
\][/tex]
5. Divide both sides by [tex]\( r \)[/tex] (assuming [tex]\( r \neq 0 \)[/tex]):
[tex]\[
r \cos^2(\theta) = 4 \sin(\theta)
\][/tex]
6. Solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{4 \sin(\theta)}{\cos^2(\theta)}
\][/tex]
The equation in polar coordinates is:
[tex]\[
r = \frac{4 \sin(\theta)}{\cos^2(\theta)}
\][/tex]
So, the given equation [tex]\( x^2 = 4y \)[/tex] in polar coordinates is represented as:
[tex]\[
r = 4 \frac{\sin(\theta)}{\cos^2(\theta)}
\][/tex]
