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

x² = 6y

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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 \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) into the given equation \(x^2 = 6y\) to rewrite it in terms of \(r\) and \(\theta\): \((r \cos{\theta})^2 = 6 (r \sin{\theta})\)

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

Since \(r\) appears on both sides, and \(r = 0\) is a trivial solution, divide both sides by \(r\) (assuming \(r \neq 0\)) to isolate \(r\): \(r \cos^2{\theta} = 6 \sin{\theta}\)

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

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Rectangular and Polar Coordinate Systems

Rectangular coordinates (x, y) represent points on a plane using horizontal and vertical distances, while polar coordinates (r, θ) represent points by their distance from the origin and 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 for conversion are x = r cos θ and y = r sin θ. These allow substitution of rectangular variables with polar expressions, enabling the transformation of equations from rectangular to polar form.

Expressing r in Terms of θ

After substituting x and y with r cos θ and r sin θ, the goal is to manipulate the equation algebraically to isolate r as a function of θ. This often involves factoring and simplifying to achieve a clear polar equation.

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.x² = 6y