Hey everyone. In this problem, we're asked to convert the equation into its rectangular form and then identify the shape of its graph. Now the equation that we're given here is r = 2 1 + cos ( θ ) . So let's jump right into our steps here and get started in taking this into its rectangular form. Now remember that we want to get our cos θ, r sin θ, or r2. So in order to do that here since I have a fraction, I'm going to go ahead and eliminate that fraction by multiplying both sides by my denominator. In this case, 1+cos(θ). Now multiplying that on both sides it will cancel on that right side, and I'm going to go ahead and distribute that r to give me r + rcos(θ). And all of this is equal to 2. Now from here, I see that I have this rcos(θ), which is great. But I also have this r, which is not typically something that we want to have. Usually, we want an r2, an r sin θ, or rcos θ. Now this is a sort of unique instance in which we actually want to keep that r as is because we already have rcos θ, so anything else that we do to this equation would affect that. So instead, we're going to go ahead and move on to step 2 and just replace what we have. But how are we going to replace that r? Well, we know that x2+y2 is equal to r2. So what if I just take the square root on both sides? Then I know that r is equal to the square root of x2 + y2. So here that's what we're going to replace our r with, the square root of x2 + y2. Then I have that rcos θ, which I can replace with just x. So I have the square root of x2 + y2 plus x is equal to 2. So I've replaced everything that I need to, and I can go ahead and move on to step number 3 where we're going to rewrite this equation in its sort of standard form. Remember that there's not always just one variable that we're going to be solving for when taking equations from polar to rectangular form. And this can be sort of tricky, but let's go ahead and look at our strategies here. Now if we see a square root, we should eliminate that square root, which we do have a square root right here. But since I have this term here as well, I'm going to go ahead and bring that to the other side to make squaring much easier. Then I have the square root of x2 + y2, and that's equal to 2 - x. Now from here, I can go ahead and square both sides, which will allow me to cancel that radical out. Now I'm just left with x2 + y2 on the left, and 4 - 4x + x2 on the right. I'm going to go ahead and multiply that out, which will give me 4 - 4x + x2. So where can we go from here? Well, I see that I have an x2 and an x2 on both sides, letting me cancel that out, leaving me with y2 is equal to 4 - 4x. Now this might not immediately look familiar because it's not a super common equation that you'll see, but this is actually the equation of a horizontal parabola. So we have taken it down to its sort of standard form, where we can recognize what type of equation that it is seeing that we have this y2 is equal to 4 - 4x. This is a horizontal parabola. Let me know if you have any questions here. Thanks for watching.
Table of contents
- 0. Fundamental Concepts of Algebra3h 29m
- 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 Angles39m
- 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 Matrices1h 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
Convert Equations Between Polar and Rectangular Forms
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