Graph r 2 = 9 sin 2 θ r^2=9\sin2\theta r 2 = 9 sin 2 θ

In this problem, we're asked to graph the equation r squared equals nine times the sine of two theta. Now this is a graph of a lemniscate because I know that when I see an r squared value that is the equation of a lemniscate. Now starting with our first step here, we want to find out where that first petal is. Now Now we want to get our value of a, which remember this value at the beginning of your equation is a squared. So we need to take that a squared, in this case nine, and take the square root of it to get three, which is our a value here. So I know that that first petal is going to land at an r value of three, but I need to determine where theta is. Now remember that whenever we're dealing with a positive sine function, so I have positive a squared sine of two theta, that tells me that theta is going to be a value of pi over four. So I can go ahead and plot this first petal at three pi over four, which will land me right here in this first quadrant. Now from here, I just want to take this petal and reflect it over the pole. So reflecting this point over the pole, I end up right out here in quadrant three. Now I can just connect these with a smooth continuous curve knowing that this Lemnis Gate looks similar to an infinity symbol. Kind of taking these petals or propellers out and graphing them like this. Now we've graphed r squared equals nine times the sine of two theta a lemniscate. Thanks for watching, and let me know if you have questions.

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Calculus

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Graph $r^2=9\sin2\theta$

Graph of r^2 = 9sin(2θ) showing a polar curve with two loops.

Verified step by step guidance

Step 1: Recognize the given polar equation r² = 9sin(2θ). This equation represents a polar curve, specifically a lemniscate, which is a figure-eight shape.

Step 2: Analyze the symmetry of the curve. Since the equation involves sin(2θ), the curve is symmetric about the origin and has two loops. The factor '2θ' indicates that the curve completes one cycle for every π radians.

Step 3: Determine the maximum value of r². The maximum value of sin(2θ) is 1, so r² = 9 when sin(2θ) = 1. This means the maximum radius r is √9 = 3.

Step 4: Plot the curve by calculating r for various values of θ. For example, when θ = 0, π/4, π/2, etc., compute r using r² = 9sin(2θ). Note that r can be positive or negative, depending on the sign of sin(2θ).

Step 5: Compare the plotted curve to the provided graphs. The correct graph will show a lemniscate with two loops, symmetric about the origin, and extending to a maximum radius of 3. Based on the images, the correct graph is the one with two loops aligned along the vertical axis.