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 9 times the sine of 2 theta. Now this is a graph of 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 wanna 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 9, and take the square root of it to get 3, which is our a value here. So I know that that first petal is going to land at an r value of 3, 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 2 theta, that tells me that theta is going to be a value of pi over 4. So I can go ahead and plot this first petal at 3 pi over 4, 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 3. Now I can just connect these with a smooth continuous curve knowing that this Lemnis Gate looks similar to an infinity symbol. I'm taking these petals or propellers out and graphing them like this. Now we've graphed r squared equals 9 times the sine of 2 theta a lemniscate. Thanks for watching, and let me know if you have questions.

Table of contents

Skip topic navigation

0. Fundamental Concepts of Algebra3h 32m
1. Equations and Inequalities3h 27m
2. Graphs1h 43m
3. Functions & Graphs2h 17m
4. Polynomial Functions1h 54m
5. Rational Functions1h 23m
6. Exponential and Logarithmic Functions2h 28m
7. Measuring Angles40m
8. Trigonometric Functions on Right Triangles2h 5m
9. Unit Circle1h 19m
10. Graphing Trigonometric Functions1h 19m
11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m
12. Trigonometric Identities 2h 34m
13. Non-Right Triangles1h 38m
14. Vectors2h 25m
15. Polar Equations2h 5m
16. Parametric Equations1h 6m
17. Graphing Complex Numbers1h 7m
18. Systems of Equations and Matrices3h 6m
19. Conic Sections2h 36m
20. Sequences, Series & Induction1h 15m
21. Combinatorics and Probability1h 45m
22. Limits & Continuity1h 49m
23. Intro to Derivatives & Area Under the Curve2h 9m

15. Polar Equations

Graphing Other Common Polar Equations

Precalculus

15. Polar Equations

Graphing Other Common Polar Equations

Previous problemNext problem

Struggling with Precalculus?

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.

Previous problemNext problem

Verified step by step guidance

The given equation is $ r^2 = 9 \sin 2\theta $. This is a polar equation that represents a type of graph known as a 'lemniscate'.

To graph this equation, we need to understand the behavior of $ \sin 2\theta $. The function $ \sin 2\theta $ oscillates between -1 and 1 as $ \theta $ varies from 0 to $ 2\pi $.

The equation $ r^2 = 9 \sin 2\theta $ implies that $ r $ can be positive or negative, depending on the value of $ \sin 2\theta $. When $ \sin 2\theta $ is positive, $ r $ will be real, and when $ \sin 2\theta $ is negative, $ r $ will be imaginary, which means there is no graph for those values.

The graph will have symmetry about the origin because $ r^2 $ is involved, meaning the graph will look the same in all four quadrants.

The graph of $ r^2 = 9 \sin 2\theta $ will form a figure-eight shape, known as a lemniscate, centered at the origin. The maximum value of $ r $ will be $ \sqrt{9} = 3 $ when $ \sin 2\theta = 1 $.