Hey everyone. In this problem, we're asked to graph the equation \( r = 2 \times \sin(3\theta) \). This is the equation of a rose of the form \( r = a \sin(n\theta) \). So let's go ahead and jump right into our steps here.

For our first step, we're going to be looking at our value of \( n \), which in this case is 3. Remember, \( n \) is the number that comes before \( \theta \) in your argument. Since 3 is an odd number, that tells us we're going to have \( n \) petals. In this case, \( n \) is 3, so we're going to have 3 petals.

Moving on to step 2, we want to actually plot our first petal. We know that we'll have 3 of them when we end up with our final graph, but we want to start by plotting that first one. In plotting our first petal, we need to look at our value of \( a \), which, looking back at my equation, I know is 2. \( A \) is the number that's at the front of your rose's equation. So I have an \( a \) value of 2, and I want to figure out where \( \theta \) is going to be. Since I have a sine function in my equation, \( \theta \) is going to be equal to \( \frac{\pi}{2} \times n \). So we have to do a little calculation here, taking \( \pi \) and dividing it by \( 2 \times n \) which we know is 3 from step 1. This leaves me with an angle of \( \frac{\pi}{6} \), so my first petal is going to be graphed at \( 2\pi/6 \). Coming up to my angle \( \frac{\pi}{6} \) and going out to 2 I know that my first petal will be right here.

Now we can find our other petals by figuring out how far they are spaced out from each other and plotting all of them. I know that my first petal is located at \( 2\pi/6 \), but I know that I have 2 other petals, so I need to figure out where those are. In order to figure out how far your petals are spaced apart, you're going to take \( 2\pi \) divided by the number of your petals, which in this case is 3. So these petals will be separated by \( \frac{2\pi}{3} \) radians. To determine where our next petal is, I need to take \( \frac{\pi}{6} \) and add \( \frac{2\pi}{3} \). When I do that, I'm going to end up getting \( \frac{5\pi}{6} \). All of my petals are the exact same length, so I can go ahead and fill in all of these \( r \) values as that \( a \) value of 2. I can plot my second petal at \( \frac{25\pi}{6} \), which will end up being right out here on my graph. Then I have one other petal to account for, so I'm going to add another \( \frac{2\pi}{3} \) to that \( \frac{5\pi}{6} \). If you do this calculation, you're going to end up getting \( \frac{3\pi}{2} \). So I can plot my last petal at \( \frac{23\pi}{2} \), which will be right here.

We want to connect this with a smooth, continuous curve. Remember that for roses, you're always going to go through the center; that's where your petals stem out from. So, we can go ahead and connect these with smooth, continuous curves that look like petals because this is a rose. We're going to try our best to make these curves smooth, but if you don't get them exactly perfect, that's okay. This is the general shape of our graph for the rose \( r = 2 \times \sin(3\theta) \). Let me know if you have any questions.