Graph r = 3 cos 4 θ r=3\cos4\theta r = 3 cos 4 θ

In this problem, we're asked to graph the equation r equals three times the cosine of four theta. Now this is the equation of a rose of the form r equals a times the cosine of n theta. Now the first thing we wanna do here is determine how many petals our rose is going to have by looking at our value of n. Now here n is four, the number right before theta in our argument. Now this is an even value which tells us we're going to have two n petals. So I'm gonna multiply two times that n value of four to get that I have eight total petals on this rose. Now moving on to step two, we actually want to plot our very first petal. Now here we want to use our a value, which is three, the thing at the very beginning of our equation. So I know that this will be located at three and then some value of theta, and we determine that value of theta by looking at what trig function we have. Here we have a cosine function or equation, so theta will be equal to zero at our first petal. So I can go ahead and plot this petal at three, zero, which will end up being right here. Now from here, we want to go ahead and figure out where all of our other petals are. We have eight total petals, so it remains to find the remaining seven. So my first petal is again at three, zero, which we have already plotted, and we can figure out how far each of our petals are spaced apart by taking two pi divided by the number of our petals. Now we have eight total petals here, so if I divide two pi by eight, this simplifies to pi over four radians. Now I know that all of my petals will have the exact same exact distance, and it remains to look at all of our theta values. Now here, it's going to be much more simple to just do this visually rather than having to write out all of these values and add pi over four seven more times. Because we're starting at zero here, it's easy to see that our next petal will be located at an angle of pi over four. Then adding another angle of pi over four, we're gonna end up at pi over two, then three pi over four, then pi, then five pi over four, then three pi over two, and finally seven pi. Now you can go ahead and count these petals, but you'll see that there are eight of them. And I can go ahead and write out these angles for you just in case. Pi over four, pi over two, three pi over four, pi, five pi over four, three pi over two, and seven pi over four. Each of these are increasing by increments of pi over four radians. Now we can go ahead and move on to our final step, connecting all of these with a smooth and continuous curve. Now remember that these look like petals so we're going to connect all of them out like this going back to that center pole every time. Now it can be difficult to accurately sketch roses just because there are so many petals that you're trying to make the same size, But generally, you should aim to have all of your petals be roughly the same size, but definitely always go out to the same length and be evenly spaced by the amount that you found. Now that we know how to graph this rose here at r equals three times the cosine of four theta, thanks for watching and I'll see you in the next one.

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Calculus

16. Parametric Equations & Polar Coordinates

Polar Coordinates

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Multiple Choice

Graph $r=3\cos4\theta$

Graph of r=3cos(4θ) showing a four-leaf rose in polar coordinates.

Graph of r=3cos(4θ) depicting a four-leaf rose with a different shape.

Graph of r=3cos(4θ) illustrating a six-leaf rose in polar coordinates.

Graph of r=3cos(4θ) showing a complex eight-leaf rose pattern.

Verified step by step guidance

Step 1: Recognize the polar equation given, r = 3cos(4θ). This equation represents a rose curve in polar coordinates. The number of petals is determined by the coefficient of θ inside the cosine function.

Step 2: Analyze the coefficient of θ, which is 4. If the coefficient is even, the rose curve will have 2n petals, where n is the coefficient. Therefore, this curve will have 8 petals.

Step 3: Understand the amplitude of the cosine function, which is 3. This determines the maximum radius of the petals. Each petal will extend outward to a maximum distance of 3 units from the origin.

Step 4: Compare the graphs provided. The correct graph should depict a rose curve with 8 petals, each reaching a maximum radius of 3 units. The symmetry of the graph should align with the periodic nature of the cosine function.

Step 5: Select the graph that matches the description of an 8-petal rose curve with the correct amplitude and symmetry. Based on the images provided, the fourth graph is the correct representation of r = 3cos(4θ).