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

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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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Watch out for the human traffickers at hivefest. You wont know it until its too late. STAY AWAY! Beware, traffickers can be women or men! They will act nice until they dont. There is human trafficking going on around this type of crypto. I have witnessed it. They literally have attempted my murder and are trying to kill me with V2K and RNM. Five years this has been happening to me, it started here, around people that are still here. Homeland security has done nothing at all, they are not here to protect us. Dont we pay them to stop shit like this? The NSA, CIA, FBI, Police and our Government has done nothing. Just like they did with the Havana Syndrome, nothing. Patriot Act my ass. The American government is completely incompetent. The NSA should be taken over by the military and contained Immediately for investigation. I bet we can get to the sources of V2K and RNM then. https://ecency.com/fyrstikken/@fairandbalanced/i-am-the-only-motherfucker-on-the-internet-pointing-to-a-direct-source-for-voice-to-skull-electronic-terrorism ..... https://ecency.com/gangstalking/@acousticpulses/electronic-terrorism-and-gaslighting--if-you-downvote-this-post-you-are-part-of-the-problem if you run into one of them you may want to immediately shoot them in the face. 187, annihilate, asphyxiate, assassinate, behead, bleed, bludgeon, boil, bomb, bone, burn, bury, butcher, cap, casket, choke, chop, club, crucify, crush, curb, decapitate, decimate, deflesh, demolish, destroy, devein, disembowel, dismember, drown, electrocute, eliminate, end, euthanize, eviscerate, execute, explode, exterminate, extinguish, finish, fry, grind, guillotine, gut, hack, hang, hit, ice, implode, incinerate, kill, liquidate, lynch, massacre, maul, microwave, mutilate, neutralize, obliterate, off, pop, poison, punnish, quarter, ruin, shank, shock, shoot, shred, skin, slay, slaughter, smoke, smother, snipe, snuff, squish, stab, strangle, stone, suffocate, suicide, SWAT, swing, terminate, torture, terrorize, whack, waste, wreck. You better fucking kill me.

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