To convert the Cartesian equation [tex]\( y = 7x^2 \)[/tex] to a polar equation, we need to follow these steps:
1. Recall the relationships between Cartesian coordinates [tex]\((x, y)\)[/tex] and polar coordinates [tex]\((r, \theta)\)[/tex]:
[tex]\[
x = r \cos(\theta)
\][/tex]
[tex]\[
y = r \sin(\theta)
\][/tex]
2. Substitute [tex]\( x = r \cos(\theta) \)[/tex] and [tex]\( y = r \sin(\theta) \)[/tex] into the Cartesian equation [tex]\( y = 7x^2 \)[/tex]:
[tex]\[
r \sin(\theta) = 7 (r \cos(\theta))^2
\][/tex]
3. Simplify the equation:
[tex]\[
r \sin(\theta) = 7 r^2 \cos^2(\theta)
\][/tex]
4. Divide both sides of the equation by [tex]\( r \)[/tex] (assuming [tex]\( r \neq 0 \)[/tex]):
[tex]\[
\sin(\theta) = 7 r \cos^2(\theta)
\][/tex]
5. Solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{\sin(\theta)}{7 \cos^2(\theta)}
\][/tex]
Therefore, the polar equation corresponding to the Cartesian equation [tex]\( y = 7 x^2 \)[/tex] is:
[tex]\[
r(\theta) = \frac{\sin(\theta)}{7 \cos^2(\theta)}
\][/tex]
