Hey, everyone. In this problem, we're asked to graph the equation r=1+2cosθ. Now, I know that this is a graph of a limaçon because I have addition happening and my a and b values are not equal to each other, therefore meaning that this cannot be a cardioid. Now, keeping in mind that this is the graph of a limaçon, let's go ahead and dive into our steps here. For our first step, we want to look at our a and b values. Now, I have an a=1, a b=2. Seeing that a is less than b, that tells me that my limaçon will have an inner loop rather than a dimple. If our limaçon has an inner loop, it also means that it will have a 0, which we can go ahead and plot on our graph here, right at our pole. Now, we can move on to step number 2 and determine our symmetry. Because our equation contains a cosine function, that tells us that our graph will be symmetric about the polar axis. This is something we want to keep in mind as we proceed to step 3 and actually start plotting some points. We want to plot our points using our quadrantal angles. So doing that here, I'm going to go ahead and plug in 0 for θ in my original equation, which is 1+2cosθ, in this case 0. The cosine of 0 is 1, so this ends up being 3. So I'm going to plot my first point at 3, which is right here on that polar axis. Now, plugging in π/2, I get 1+2cosπ2. The cosine of π/2 is 0, so this ends up just being 1. I can go ahead and plot that point at 1 π/2, which will be right here. Now, remembering my symmetry here means that I can just reflect this point over the polar axis, giving me another point at my quadrantal angle of 3π/2 with an r value of 1. Now, we just need to plug in that π to get one final point here to get the complete picture of our graph. Now when I plug in π here, I get 1+2cosπ. The cosine of π is -1. So this will end up giving me 1 minus 2, which will give me -1. Now, remember when we have a negative r value, that means instead of counting out towards our angle, we want to count in the opposite direction. So that ends up giving me a point right here again on that polar axis. Now from here, we want to connect this with a smooth and continuous curve. Remembering back in step 1, we said that this limaçon has an inner loop. So what exactly does that look like? Well, it looks almost identical to the other limaçon, but now there's an extra little circle or loop in the middle. This point shows us what our inner loop is going to be, the point closest to the pole. So I have that inner loop right there. And then my outer loop, the rounded bottom of my limaçon, is going to be out here at the furthest point away from the pole, still having a curved line rounding around to that bottom point. This is the graph of my limaçon for the equation r=1+2cosθ. Thanks for watching, and let me know if you have questions.
Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles39m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
9. Polar Equations
Graphing Other Common Polar Equations
Video duration:
2mPlay a video:
Related Videos
Related Practice
Graphing Other Common Polar Equations practice set
- Problem sets built by lead tutorsExpert video explanations