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 the Hive blockchain: https://peakd.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: 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: https://youtu.be/usSors49Gdw .

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