Q. 14 - Calculus

Question

Using polar coordinates, the graphs of the equations$r = \cos k\theta$ and $r = \sin k\theta$are roses with k petals when $k \ge 3$ is an odd integer. What are the polar graphs of these equations when$k = 1$? What are the graphs of these equations when k is a negative odd integer?

Step-by-Step Solution

Answer

The graph of the curves is a circle when $k=1$

The graph of the curves is a roses having $\left|k\right|$ number of petals if $k$ is a negative odd integers.

Step 1. Given information

Polar curves:

$$\begin{gathered} r=\cos k\theta \\ r=\sin k\theta \end{gathered}$$

$$\begin{gathered} k=1 \\ k\ge 3 \end{gathered}$$

Step 2. Graphs of curves when k = 1 .

When $k=1$ then the graph of $r=\cos \theta$ and graph of $r=\sin \theta$

Step 3. Graph of the curves when k is negative odd integers.

When $k$ is negative odd integers then the graph of the given curves are also a rose having $\left|k\right|$ number of petals.

For example, if $k=-5$ then graph of $r=\cos \left(-5\theta \right)$ and $r=\sin \left(-5\theta \right)$ are drawn as: