In this discussion, we explore the conversion of equations from rectangular to polar form, a crucial skill in understanding the relationship between Cartesian coordinates and polar coordinates. The polar coordinates are defined using the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \), where \( r \) is the radius and \( \theta \) is the angle.

For the first example, we start with the equation \( y = x^2 \). By substituting \( y \) with \( r \sin \theta \) and \( x \) with \( r \cos \theta \), we rewrite the equation as:

\( r \sin \theta = (r \cos \theta)^2 \)

Squaring the right side gives us:

\( r \sin \theta = r^2 \cos^2 \theta \)

To isolate \( r \), we can divide both sides by \( r \) (assuming \( r \neq 0 \)), leading to:

\( \sin \theta = r \cos^2 \theta \)

Rearranging this, we find:

\( r = \frac{\sin \theta}{\cos^2 \theta} \

In the second example, we convert the equation \( 4xy = 2 \). Substituting \( x \) and \( y \) gives us:

\( 4(r \cos \theta)(r \sin \theta) = 2 \

This simplifies to:

\( 4r^2 \cos \theta \sin \theta = 2 \

Dividing both sides by 2 results in:

\( 2r^2 \cos \theta \sin \theta = 1 \

Recognizing that \( \cos \theta \sin \theta \) is part of the double angle identity, we can express it as:

\( r^2 \cdot \sin(2\theta) = 1 \

To solve for \( r^2 \), we divide both sides by \( \sin(2\theta) \), yielding:

\( r^2 = \frac{1}{\sin(2\theta)} \

Since \( \frac{1}{\sin(2\theta)} \) is equivalent to \( \csc(2\theta) \), we conclude:

\( r^2 = \csc(2\theta) \

Through these examples, we see how to effectively convert rectangular equations into polar form, utilizing trigonometric identities and algebraic manipulation to achieve the desired results.