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

x² + y² = 16

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

Substitute \(x = r \cos\theta\) and \(y = r \sin\theta\) into the given equation \(x^{2} + y^{2} = 16\).

This substitution gives the equation \((r \cos\theta)^{2} + (r \sin\theta)^{2} = 16\).

Simplify the equation by factoring out \(r^{2}\): \(r^{2}(\cos^{2}\theta + \sin^{2}\theta) = 16\).

Use the Pythagorean identity \(\cos^{2}\theta + \sin^{2}\theta = 1\) to simplify to \(r^{2} = 16\), then solve for \(r\) to express it in terms of \(\theta\).

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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.

Relationship Between Rectangular and Polar Coordinates

The key formulas connecting the two systems are x = r cos θ and y = r sin θ. Additionally, r² = x² + y². These relationships allow substitution of x and y in terms of r and θ to rewrite rectangular equations in polar form.

Converting Equations to Express r in Terms of θ

To convert a rectangular equation to polar form expressing r as a function of θ, substitute x and y with r cos θ and r sin θ, respectively, then solve algebraically for r. This process isolates r, showing how the radius varies with the angle θ.

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.x² + y² = 16