identify and sketch the graph of the polar equation. r = 4 cos(6θ)

Explanation:

Step1: Recall polar - rose curve formula

The general form of a polar rose - curve is $r = a\cos(n\theta)$ or $r=a\sin(n\theta)$. When $n$ is even, the number of petals is $2n$; when $n$ is odd, the number of petals is $n$. Here, $a = 4$ and $n = 6$ (even).

Step2: Determine the number of petals

Since $n = 6$ (even), the number of petals of the rose - curve $r = 4\cos(6\theta)$ is $2n=12$.

Step3: Find the maximum value of $r$

The maximum value of $\cos(6\theta)$ is 1. So, the maximum value of $r$ is $r_{max}=4\times1 = 4$.

Step4: Sketch the curve

We can find the values of $r$ for some key - angles: When $\theta = 0$, $r = 4\cos(0)=4$. When $\theta=\frac{\pi}{12}$, $r = 4\cos\left(6\times\frac{\pi}{12}\right)=0$. We can then plot these points and draw a smooth curve to form a 12 - petaled rose centered at the origin with a maximum radius of 4.

Answer:

The graph is a 12 - petaled rose curve with a maximum radius of 4.