Convert each rectangular equation to a polar equation that expresses r in terms of θ.
y = 3

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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 \(y = 3\), substitute \(y\) with its polar form: \(r \sin\theta = 3\).

To express \(r\) in terms of \(\theta\), isolate \(r\) by dividing both sides of the equation by \(\sin\theta\): \(r = \frac{3}{\sin\theta}\).

Note that this expression is valid for values of \(\theta\) where \(\sin\theta \neq 0\), since division by zero is undefined.

Thus, the polar equation expressing \(r\) in terms of \(\theta\) corresponding to the line \(y = 3\) is \(r = \frac{3}{\sin\theta}\).

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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 the relationship between these systems 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). These allow expressing rectangular variables x and y in terms of polar variables r and θ, enabling the transformation of equations from rectangular to polar form.

Expressing r in Terms of θ

To convert an equation like y = 3 into polar form, substitute y with r sin(θ) and solve for r. This process isolates r as a function of θ, which is the goal when expressing polar equations explicitly.

Convert each rectangular equation to a polar equation that expresses r in terms of θ.y = 3