49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.

y = x²

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Recall the relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).

Substitute \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) into the given equation \(y = x^2\) to rewrite it in terms of \(r\) and \(\theta\).

This substitution gives \(r \sin{\theta} = (r \cos{\theta})^2\).

Simplify the right side to get \(r \sin{\theta} = r^2 \cos^2{\theta}\).

To isolate \(r\), divide both sides by \(r\) (assuming \(r \neq 0\)), resulting in \(\sin{\theta} = r \cos^2{\theta}\). Then solve for \(r\) to express the equation fully in polar form.

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Cartesian and Polar Coordinate Systems

Cartesian coordinates represent points using (x, y) values on perpendicular axes, while polar coordinates use (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding the relationship between these systems is essential for converting equations between them.

Conversion Formulas Between Cartesian and Polar Coordinates

The key formulas for conversion are x = r cos(θ) and y = r sin(θ). To convert an equation from Cartesian to polar form, substitute x and y with these expressions and simplify to express the equation in terms of r and θ.

Manipulating and Simplifying Trigonometric Expressions

After substitution, the resulting equation often involves trigonometric functions like sine and cosine. Being able to manipulate and simplify these expressions is crucial to rewrite the equation clearly in polar form, such as isolating r or expressing it as a function of θ.

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.y = x²