Polar Coordinates: Example 12: Limaçons Analytical Proof: Part 1: Question A

In this video I go over further into Polar Coordinates and this time look at Example 12 Question A which revisits the Limaçons curves from Example 11 but now I take an analytical approach as opposed to the graphical approach from the previous example. Question A looks at the “loop” which I showed occurred for values of c that are larger than 1 or less than -1, i.e. the absolute value of c is greater than 1, in the Limaçon formula r = 1 + c sin ϴ. We are asked to prove this is the case, and thus I go over an VERY extensive analytical proof by first showing that the requirement for the loop is to have two angles where the coordinates are at the origin, and between them the r-values are negative. I show that this is only possible when |c| is greater than 1. This is an extremely detailed proof video, but if you follow along the whole thing you will be that much more knowledgeable about mathematics in general, so make sure to watch this video! I will be going over Question B in the next video so #StayTuned!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvRsaO7qUm_nIAzIXw

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/video-notes-polar-coordinates-example-12-limacons-analytical-proof-part-1-question-a

Related Videos:

Polar Coordinates: Example 11: Limaçons + Cochlea + Archimedean Screw: https://youtu.be/GByUVHGAMe0 Polar Coordinates: Example 10: Graphing r = sin(8ϴ/5): https://youtu.be/m-A35NDX7b0 Polar Coordinates: Graphing With Polar Curves with Desmos Calculator: https://youtu.be/xY9Be9HHRsk Polar Coordinates: Example 9: Cardioid: Part 4: https://youtu.be/R6UtQD8tUF0 Polar Coordinates: Example 9: Cardioid: Part 3: Question B: https://youtu.be/-TZHaTjHfb8 Polar Coordinates: Example 9: Cardioid: Part 2: Question A: https://youtu.be/gUYmzZ8Cfhg Polar Coordinates: Example 9: Cardioid: Part 1: Slope Formula & Trigonometric Algebra: https://youtu.be/MlIhueP36EE Polar Coordinates: Tangents to Polar Curves: https://youtu.be/DJso0gmxCBY Polar Coordinates: Symmetry: https://youtu.be/TWxforTOvxM Polar Coordinates: Example 8: Four-Leaved Rose: https://youtu.be/mTk5j0jD3dE Polar Coordinates: Example 7: Cardioid: https://youtu.be/rPErcaqNUIY Polar Coordinates: Example 6: Part 2: Polar Circle to Cartesian: https://youtu.be/biUHN-BphkE Polar Coordinates: Example 5: Straight Lines: https://youtu.be/AGSz3EVi05A Polar Coordinates: Example 4: Circle: https://youtu.be/ebsaIlXaZxs Polar Coordinates: Example 3: Cartesian to Polar Polar Coordinates: Example 2: Polar to Cartesian: https://youtu.be/mvpb9QshNNU Polar Coordinates: Cartesian Connection: https://youtu.be/HcaTYrpmGaU Polar Coordinates: Example 1: https://youtu.be/q_kpqPpoLqE Polar Coordinates: https://youtu.be/-KAdZL-N4ok Parametric Equations and Polar Coordinates: h ... https://www.youtube.com/watch?v=90XjQVkLBVY