Problem 68 - Essential Calculus

Question

Sketch a graph of the polar equation. $$ r=1+\sin \theta $$

Step-by-Step Solution

Answer

To sketch the graph of the polar equation \(r = 1+\sin\theta\), you have to plot the curve by evaluating \(r\) at various values of \(\theta\) within the range \(0 \leq \theta \leq 2\pi\). The resulting plot reveals a heart-shaped curve known as a cardioid.

Step 1: Set up the range of theta

Start by determining the values of theta within the standard range for trigonometric functions: \(0 \leq \theta \leq 2\pi\). You may want to use intervals of \(\frac{\pi}{2}\) for convenience as all usable values of the sine function are obtained within this range.

Step 2: Evaluate r

Calculate the corresponding values of \(r\) by substituting the chosen values of theta into the equation \(r = 1+\sin{\theta}\). This step gives us the distance from the origin of each point on the curve.

Step 3: Plot the curve

Now is the time to plot the curve. Each point on the curve is represented by a direction and a distance from the origin (0,0) in polar coordinates. Use the values from steps 1 and 2 to plot these points, and then connect these points smoothly to form the curve. You'll notice the unique pattern of this curve, which is sometimes called a 'cardioid' due to its heart-like shape.