Problem 67 - Algebra and Trigonometry

Question

Use a graphing utility to graph the polar equation. $$r=2+4 \sin \theta$$

Step-by-Step Solution

Answer

The converted Cartesian equation, once simplified, can then be graphed in a graphing utility to represent the polar equation \( r = 2 + 4 \sin \theta \) visually. Though the graph isn't drawn here, the main steps include conversion to Cartesian coordinate system, simplification of derived equation, and graphing the final equation using a suitable tool.

Step 1: Convert the polar equation to Cartesian

We know that \(r = \sqrt{x^2 + y^2}\) and \(\sin \theta = \frac{y}{r}\). Replace r and sin \theta in the given equation, \(r = 2 + 4 sin \theta\), with these relations to convert into Cartesian coordinates. Thus, we get \( \sqrt{x^2 + y^2} = 2 + 4 \frac{y}{\sqrt{x^2 + y^2}}\).

Step 2: Simplify the equation

To simplify the equation, square both sides to get rid of the square roots: \(x^2 + y^2 = 4 + 16y + 4y^2\). Now, rearrange the terms and simplify to achieve a standard form of equation.

Step 3: Graph the equation

Use any available graphing utility and plot the equation derived in step 2. It will effectively visualize the original polar equation.