Question

How do you graph the given function $r=5\sin 5\theta$ ?

Answer 1

Hint: We start solving the problem by finding the maximum and minimum values of the function using the fact that the values of the function $a\sin bx$ lies in the interval $[-a,a]$ . We then make use of the fact that the maximum value of the function $a\sin bx$ occurs at $bx=(4n+1){\displaystyle \frac{\pi }{2}}$ , $n\in \mathbb{Z}$ to find the values of $\theta$ at which we get the maximum values of the given function. We then make use of the fact that that the minimum value of the function $a\sin bx$ occurs at $bx=(4n-1){\displaystyle \frac{\pi }{2}}$ , $n\in \mathbb{Z}$ to find the values of $\theta$ at which we get the minimum values of the given function. We then make use of the fact that that the function $a\sin bx$ is equal to 0 at $bx=n\pi$ , $n\in \mathbb{Z}$ to find the values of $\theta$ at which the given function is equal to 0. We then plot all the obtained points to get the required graph.

Complete step by step answer:
According to the problem, we are asked to plot the given function $r=5\sin 5\theta$ on a graph.
So, we have the function $r=5\sin 5\theta$ ---(1).
We know that the values of the function $a\sin bx$ lies in the interval $[-a,a]$ .
So, the values of the function $r=5\sin 5\theta$ lies in the interval $[-5,5]$ .
We know that the maximum value of the function $a\sin bx$ occurs at $bx=(4n+1){\displaystyle \frac{\pi }{2}}$ , $n\in \mathbb{Z}$ .
So, the maximum value of the function $r=5\sin 5\theta$ occurs at $5\theta =(4n+1){\displaystyle \frac{\pi }{2}}\iff \theta =(4n+1){\displaystyle \frac{\pi }{10}}$ , $n\in \mathbb{Z}$ ---(1).
We know that the minimum value of the function $a\sin bx$ occurs at $bx=(4n-1){\displaystyle \frac{\pi }{2}}$ , $n\in \mathbb{Z}$ .
So, the minimum value of the function $r=5\sin 5\theta$ occurs at $5\theta =(4n-1){\displaystyle \frac{\pi }{2}}\iff \theta =(4n-1){\displaystyle \frac{\pi }{10}}$ , $n\in \mathbb{Z}$ ---(2).
We know that the function $a\sin bx$ is equal to 0 at $bx=n\pi$ , $n\in \mathbb{Z}$ .
So, the value of the function $r=5\sin 5\theta$ is 0 at $5\theta =n\pi \iff \theta ={\displaystyle \frac{n\pi }{5}}$ , $n\in \mathbb{Z}$ ---(3).
Now, let us plot the points obtained at all the points obtained from equations (1), (2), and (3) to get the required graph which is as shown below.

Note:
We should not confuse the general solution for the minimum and maximum values of the function $a\sin bx$ as this will give the wrong graph for us. Whenever we get this type of problem, we first find the points at which we get the minimum, maximum, and 0 to plot the graph. Similarly, we can expect problems to plot the graph of the function $r=3\cos 6\theta +3\sin 4\theta$.