Problem 70 - Algebra and Trigonometry

Question

Use a graphing utility to graph the polar equation. $$r=\cos \frac{3}{2} \theta$$

Step-by-Step Solution

Answer

The graphing process will be easier if the equation is simplified into a Cartesian format and then graphed using a proper graphing tool.

Step 1: Transformation of Polar Equation to Cartesian Equation

To transform the given equation \(r=\cos (\frac{3}{2} \theta)\) into Cartesian form, use the relations \[x=r\cos(\theta)\] and \[y=r\sin(\theta).\] Since \(r^2 = x^2 + y^2\) and \(\theta = \arctan\left(\frac{y}{x}\right)\), replace \(r\) and \(\theta\) by \(x\) and \(y\) in the equation.

Step 2: Simplify Equation

After substitution we get the equation in Cartesian form. Then simplify the equation as much as possible.

Step 3: Graphing the Equation

After getting a simplified Cartesian equation, use a graphing utility to sketch the graph. Note that some graphing utilities allow for graphing in polar coordinates directly, in that case, the transformation steps can be skipped.