In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. x = 7

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

Given the rectangular equation \(x = 7\), substitute \(x\) with \(r \cos{\theta}\) to get \(r \cos{\theta} = 7\).

To express \(r\) in terms of \(\theta\), isolate \(r\) by dividing both sides of the equation by \(\cos{\theta}\), resulting in \(r = \frac{7}{\cos{\theta}}\).

Recognize that \(\frac{1}{\cos{\theta}}\) is the secant function, so the polar equation can also be written as \(r = 7 \sec{\theta}\).

This polar equation expresses \(r\) explicitly in terms of \(\theta\), completing the conversion from rectangular to polar form.

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Rectangular and Polar Coordinate Systems

Rectangular coordinates represent points using (x, y) values on a Cartesian 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 the relationship between these systems is essential for converting equations.

Conversion Formulas Between Coordinates

The key formulas for conversion are x = r cos(θ) and y = r sin(θ). To convert from rectangular to polar, express x and y in terms of r and θ, then manipulate the equation to isolate r as a function of θ.

Expressing r in Terms of θ

After substituting x = r cos(θ) into the given equation, solve for r to express it explicitly as a function of θ. This step is crucial to rewrite the rectangular equation in polar form, enabling analysis or graphing in polar coordinates.