Q. C - Calculus

Question

When $k \ge 2$ is an integer, the polar graph of $r = \sin k\theta \mathrm{or} r = \cos k\theta$ is a rose. (a) How many petals does the rose have when k is an integer? (b) What can you say about the symmetries of either$r = \sin k\theta or r = \cos k\theta$ when k is rational? Use a graphing calculator or a computer algebra system to graph several cases before answering the question. (c) What can you say about the polar graph of  $r = \sin k\theta or r = \cos k\theta$when k is irrational? 

Step-by-Step Solution

Answer

(a). There are 2k and k number of rose petals when k is an even and odd integer.

(b). The graph is symmetric when k is an integer.

(c). The polar graph has infinite number of petals when k is irrational 

Step 1: Given information

$r = \sin k\theta or r = \cos k\theta$

Part (a) Step 1: Number of rose petals when k is an integer.

The graph of the given curves is,

It is observed that when k is even, there are 2k petals on the rose, and when k is odd, there are only k petals. This pattern emerges if k is any integer, including if k was negative. 

Part (b) Step 1: Symmetry

For $k=3$ the graph is,

When $k$ is odd there will be $k-1$ symmetry.

For $k=6$

When $k$ is even there are $k$ symmetry

Step 3: Graph when k is an irrational number.

When $k$ is an irrational number, then the graph is as below

The graph for different irrational numbers:

There are infinite numbers of petals when k is an irrational numbers.