Question

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

Short 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-by-step solution

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.

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.

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.