In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.

y² = 6x

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

Substitute \(x\) and \(y\) in the given equation \(y^2 = 6x\) using the polar forms: \((r \sin{\theta})^2 = 6 (r \cos{\theta})\)

Simplify the equation: \(r^2 \sin^2{\theta} = 6r \cos{\theta}\)

Since \(r\) can be zero, consider dividing both sides by \(r\) (assuming \(r \neq 0\)) to isolate \(r\): \(r \sin^2{\theta} = 6 \cos{\theta}\)

Finally, solve for \(r\) in terms of \(\theta\): \(r = \frac{6 \cos{\theta}}{\sin^2{\theta}}\)

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Rectangular and Polar Coordinate Systems

Rectangular coordinates represent points using (x, y) on a plane, 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 how these systems relate is essential for converting equations between them.

Conversion Formulas Between Rectangular and Polar Coordinates

The key formulas are x = r cos θ and y = r sin θ, which allow substitution of rectangular variables with polar expressions. These formulas enable rewriting equations from rectangular form into polar form by expressing x and y in terms of r and θ.

Algebraic Manipulation to Express r in Terms of θ

After substituting x and y with their polar equivalents, algebraic techniques are used to isolate r on one side of the equation. This step is crucial to express r explicitly as a function of θ, completing the conversion to a polar equation.

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.y² = 6x