Graphing Polar Equations | Calculus 2 Lesson 47 - JK Math

How to Graph Polar Equations (Calculus 2 Lesson 47) In this video we learn how to graph polar equations. This includes polar equations that represent basic polar graphs such as lines and circles, as well as more complex and special polar graphs such as limacons with an inner loop, cardioids, limacons with a dimple, convex limacons, rose curves, and lemniscates. We look at a method that will allow us to graph any polar equation by plotting points and making use of symmetry, and then we look at rules to help us quickly sketch special polar graphs by recognizing the form of a polar equation. ⬇️ 📝 Download My Free Blank Polar Graphs Here: https://www.jkmathematics.com/blank-polar-coordinates-worksheet 🖥️ Join My Membership Site: https://www.jkmathematics.com/plus This video series is designed to help students understand the concepts of Calculus 2 at a grounded level. No long, boring, and unnecessary explanations, just what you need to know at a reasonable and digestible pace, with the goal of each video being shorter than the average school lecture! Calculus 2 requires a solid understanding of calculus 1, precalculus, and algebra concepts and techniques. This includes limits, differentiation, basic integration, factoring, equation manipulation, trigonometric functions, logarithms, graphing, and much more. If you are not familiar with these prerequisite topics, be sure to learn them first! Video Chapters: 0:00 What are Polar Graphs? (Intro) 2:10 Graphs of r=a & θ=a 4:50 Graphs of r=a*secθ & r=a*cscθ 8:19 Graphs of r=a*cosθ & r=a*sinθ 10:44 Graphing Using Points & Symmetry 12:37 Example: Graphing r=1+cosθ (Points & Symmetry) 26:08 Limacons With Inner Loop 30:36 Cardioids 32:25 Limacons With Dimple 34:33 Convex Limacons 37:11 Rose Curves (cosθ) 40:34 Rose Curves (sinθ) 44:58 Lemniscates 47:56 Outro 📝 Examples Video: https://youtu.be/j79xtgVxTQM ⏩ Next Lesson: https://youtu.be/3WQF8nWtt3M 📺 Calculus 2 Playlist: https://youtube.com/playlist?list=PLHdRLeAbIZE7KQ922piyRIDAj30XYcOIu 📺 Calculus 1 Playlist: https://youtube.com/playlist?list=PLHdRLeAbIZE6krGDQthaQRpLZFLD4ob54 🌐 Visit My Website: https://www.jkmathematics.com Corrections: 16:17 I accidentally wrote the identity wrong here. It should be cosAcosB + sinAsinB. You want to add the terms, not subtract. Thankfully the final answer is not effected by this mistake since adding/subtracting 0 produces the same result. I got lucky, but that will not always be the case when checking symmetry! 46:21 I slightly misspoke when I said to rotate the lemniscates by 180° or π radians when the coefficient is negative. In reality you only rotate by 90° or π/2 radians (just like I show visually). What I meant to say is that each angle θ you use in the equation should be increased by π (or 180°) to adjust for the negative coefficient. This is because sine and cosine have the properties of -sin(θ)=sin(θ+π) and -cos(θ)=cos(θ+π). This is important because then if you were to solve for r by taking the square root of both sides of the polar equation, you don't have to take the square root of a negative number. The negative coefficient can be removed by adding π to the angle. ⚡️Math Products I Recommend⚡️ Graphing Calculator: https://amzn.to/3XuUcB9 Non-Graphing Calculator: https://amzn.to/46hHJ7E Financial Calculator: https://amzn.to/3CRe9s4 Graphing Paper: https://amzn.to/46mAfAr College Rule Paper: https://amzn.to/3NOJaTN My Favorite Pencil: https://amzn.to/3NOFMIB My Favorite Erasers: https://amzn.to/3PumUQB Calculus Workbook: https://amzn.to/3r8SnNP ⚡️Textbooks I Use⚡️ Calculus 1 & 2: https://amzn.to/3PzfI5F Calculus 3: https://amzn.to/46rSy76 Financial Mathematics: https://amzn.to/3ppIVW0 ⚡️My Recording Equipment⚡️ iPad Air: https://amzn.to/3phEug1 Apple Pencil (Gen 2): https://amzn.to/3Nl45MM Tablet Desk Stand: https://amzn.to/4420rPh External Hardrive: https://amzn.to/44pkEyh (Commissions earned on qualifying purchases) Find me on social media: Facebook: https://www.facebook.com/jkmathematics Twitter: https://twitter.com/jk_mathematics Instagram: @jk_mathematics Found this video to be helpful? Consider giving this video a like and subscribing to the channel! Thanks for watching! Any questions? Feedback? Leave a comment! -Josh from JK Math #calculus Disclaimer: Please note that some of the links associated with the videos on my channel may generate affiliate commissions on my behalf. As an amazon associate, I earn from qualifying purchases that you may make through such affiliate links.