Question

Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as $\theta$ increases from 0 to $2\pi$. $$r=\frac{1}{1+\sin \theta }$$

Short answer

Answer: The curve of the polar equation $r=\frac{1}{1+\sin \theta }$ forms a shape that resembles a combination of a circle and a cardioid. Some key points on this curve include (1,0), (1,1), (0,1/2), (-1,1), (-1,0), (-1,-1), (0,-1/2), and (1,-1).

Step-by-step solution

Understanding Polar Coordinates

The polar coordinate system is a two-dimensional coordinate system in which each point in the plane is determined by its distance from a reference point (called the pole or origin) and the angle it makes with a reference direction (ie. the polar angle). In this system, we use the notation $(r,\theta )$ to represent the polar coordinates of a point. Here, $r$ represents the radial distance, and $\theta$ represents the polar angle.

Creating the Table

To visualize how the values of $r$ change with increasing $\theta$, we should construct a table with $\theta$ values from $0$ to $2\pi$ (increasing by $\pi /4$ as an example) and compute their corresponding $r$ values using the given equation. Afterwards, we will convert the polar coordinates into rectangular coordinates $(x,y)$ to help plot the points on a graph: $$r=\frac{1}{1+\sin \theta }$$ Table: $\begin{tabular}{c|c|c|c} \hline $\mathrm{\Theta }$ & $\sin \theta$ & $r$ & $(x,y)$ \ \hline $0$ & $0$ & $1$ & $(1,0)$ \ $\pi /4$ & $1/\sqrt{2}$ & $2/\sqrt{2}$ & $(1,1)$ \ $\pi /2$ & $1$ & $1/2$ & $(0,1/2)$ \ $3\pi /4$ & $1/\sqrt{2}$ & $2/\sqrt{2}$ & $(-1,1)$ \ $\pi$ & $0$ & $1$ & $(-1,0)$ \ $5\pi /4$ & $-1/\sqrt{2}$ & $2/\sqrt{2}$ & $(-1,-1)$ \ $3\pi /2$ & $-1$ & $1/2$ & $(0,-1/2)$ \ $7\pi /4$ & $-1/\sqrt{2}$ & $2/\sqrt{2}$ & $(1,-1)$ \ $2\pi$ & $0$ & $1$ & $(1,0)$ \ \hline \end{tabulum}$

Plotting the Points

Now that the table is complete, we can plot the points on a polar graph paper. Starting with the points at $\theta =$ 0, $\pi /4$, $\pi /2$, $3\pi /4$, $\pi$, $5\pi /4$, $3\pi /2$, $7\pi /4$, and $2\pi$, respectively. As the table shows, some key points to note are: - At $\theta =0,2\pi$: $r=1$, which makes a circle around the origin or pole. - At $\theta =\pi /2,3\pi /2$: $r=1/2$, which forms a cardioid-like shape around the origin. - At $\theta =\pi /4,3\pi /4,5\pi /4,7\pi /4$: $r=2/\sqrt{2}=\sqrt{2}$, which connects the circle and cardioid shapes.

Tracing the Curve and Labeled Points

Finally, trace the curve of the polar equation and indicate the direction in which the curve is generated as $\theta$ increases from $0$ to $2\pi$. Remember to draw arrows in the direction of increasing $\theta$ and label important points along the curve (such as $(1,0)$, $(1,1)$, $(0,1/2)$, $(-1,1)$, $(-1,0)$, etc.). By following these steps, the student will be able to understand and plot the given polar equation and its corresponding curve.