Problem 95 - Algebra and Trigonometry

Question

In converting \(r=\sin \theta\) from a polar equation to a rectangular equation, describe what should be done to both sides of the equation and why this should be done.

Step-by-Step Solution

Answer

To convert the polar equation \(r = \sin \theta\) into a rectangular equation, we substitute \(r\) by \(y / \sin \theta\) from the conversion formula, resulting in the rectangular equation \(y = \sin^2 \theta\).

Step 1: Understand Conversion Formulas

The first step is to understand the conversion formulas between polar and rectangular coordinates. We have the equations \(x = r \cos \theta\) and \(y = r \sin \theta\). Inverting the second equation yields \(r = y / \sin \theta\).

Step 2: Substitute Polar Equation

In the next step, replace \(r\) in the polar equation \(r = \sin \theta\) by \(y / \sin \theta\), obtained from the conversion formula. This gives the equation \(y / \sin \theta = \sin \theta\).

Step 3: Simplify Equation

Now, simplify this equation by multiplying both sides by \(\sin \theta\) to remove the denominator on the left side. This yields the rectangular equation \(y = \sin^2 \theta\).