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 3 times the cosine of 4 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 4, the number right before theta in our argument. Now this is an even value, which tells us we're going to have 2 n petals. So I'm gonna multiply 2 times that n value of 4 to get that I have 8 total petals on this rose. Now moving on to step 2, we actually want to plot our very first petal. Now here, we want to use our a value, which is 3, the thing at the very beginning of our equation. So I know that this will be located at 3 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 0 at our first petal. So I can go ahead and plot this petal at 30, 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 8 total petals so it remains to find the remaining 7. So my first petal is again at 30 which we have already plotted and we can figure out how far each of our petals are spaced apart by taking 2 pi divided by the number of our petals. Now we have 8 total petals here. So if I divide 2 pi by 8, this simplifies to pi over 4 radians. Now I know that all of my petals will have the exact same r value. They all go out to the 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 4 7 more times. Because we're starting at 0 here, it's easy to see that our next petal will be located at an angle of pi over 4. Then adding another angle of pi over 4, we're going to end up at pi over 2, then a 3pi over 4, then pi, then 5pi over 4, then 3pi over 2, and finally 7pi. Now you can go ahead and count these petals, but you'll see that there are 8 of them. And I can go ahead and write out these angles for you just in case. Pi over 4, pi over 2, 3 pi over 4, pi, 5pi over 4, 3pi over 2, and 7pi over 4. Each of these are increasing by increments of pi over 4 radians. Now we can go ahead and move on to our final step, connecting all of these with a smooth and continuous curve. Now 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 3 times the cosine of 4 theta, thanks for watching and I'll see you in the next one.

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15. Polar Equations

Graphing Other Common Polar Equations

Precalculus

15. Polar Equations

Graphing Other Common Polar Equations

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Multiple Choice

Graph $r=3\cos4\theta$

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Verified step by step guidance

Identify the polar equation given: $ r = 3 \cos 4\theta $. This is a polar equation representing a rose curve.

Understand the structure of rose curves: The general form $ r = a \cos n\theta $ or $ r = a \sin n\theta $ produces a rose curve with $ n $ petals if $ n $ is odd, and $ 2n $ petals if $ n $ is even.

In the given equation, $ n = 4 $, which is even, so the rose curve will have $ 2 \times 4 = 8 $ petals.

The coefficient $ a = 3 $ determines the length of each petal. Each petal will extend to a maximum radius of 3 units from the origin.

Examine the images provided: The correct graph should display a rose curve with 8 petals, each extending to a radius of 3 units. Compare the images to identify the graph that matches these characteristics.