for the following function, (a) describe the graph of the polar function, (b) state any symmetry that the…

Explanation:

Step1: Analyze the form of the polar function

The function $r = 3-5\cos(5\theta)$ is a rose - curve. The general form of a rose - curve is $r = a\pm b\cos(n\theta)$ or $r=a\pm b\sin(n\theta)$. When $n$ is odd, the rose has $n$ petals; when $n$ is even, the rose has $2n$ petals. Here $n = 5$ (odd), so the graph has 5 petals.

Step2: Check for symmetry

1. Symmetry about the polar axis ($\theta = 0$): Replace $\theta$ with $-\theta$. We get $r=3 - 5\cos(- 5\theta)$. Since $\cos(-x)=\cos(x)$, then $r = 3-5\cos(5\theta)$, which is the original function. So the graph is symmetric about the polar axis.
2. Symmetry about the line $\theta=\frac{\pi}{2}$: Replace $\theta$ with $\pi-\theta$. Then $r=3 - 5\cos(5(\pi - \theta))=3-5\cos(5\pi - 5\theta)$. Using the identity $\cos(A - B)=\cos A\cos B+\sin A\sin B$, $\cos(5\pi - 5\theta)=-\cos(5\theta)$. So $r=3 + 5\cos(5\theta)\neq3 - 5\cos(5\theta)$ (not symmetric about $\theta=\frac{\pi}{2}$).
3. Symmetry about the pole: Replace $r$ with $-r$. We get $-r=3 - 5\cos(5\theta)$ or $r=-3 + 5\cos(5\theta)\neq3 - 5\cos(5\theta)$ (not symmetric about the pole).

Step3: Find the maximum value of $|r|$

We know that $-1\leqslant\cos(5\theta)\leqslant1$. To find the maximum of $r = 3-5\cos(5\theta)$, we consider the extreme values of $\cos(5\theta)$. When $\cos(5\theta)=-1$, $r=3-5\times(-1)=3 + 5=8$.

Answer:

(a) The graph is a rose - curve with 5 petals. (b) The graph is symmetric about the polar axis ($\theta = 0$). (c) The maximum value of $|r|$ is 8.