Parametric Curves: Example 9: The Cycloid: Proof Part 3 (Notes) by mes

mathematics · @mes · Feb 6 '18 (edited)

$9.59

Parametric Curves: Example 9: The Cycloid: Proof Part 3 (Notes)

https://youtu.be/4VuYJRFAI_g

In this video I go over further into proving the parametric equations for the Cycloid curve and this time show that these equations do in fact work for all angles that the circle rotates to. In my earlier videos I derived the proof for the cases where θ is between 0 and  π/2, as well as between π/2 and π. In this video I first show that the equations still work for the values θ = 0, π/2, and pi. From there I show that it still works for all other values of θ because of the cyclical nature of the trigonometric functions in the cycloid equations. Thus as the values of θ change, the derivation for the parametric equations will mirror all our previous derivations and thus give the exact same result. This is a great example to see how we can spot patterns in the derivation of parametric equations that involve trig functions, so make sure to watch this video!

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# Example: 

Given that from my earlier videos I proved the parametric equations for the cases: 

Case 1: 0 < θ < π/2
Case 2: π/2 < θ < π

Show that in fact the parametric equations are valid for all values of θ.

![](https://steemitimages.com/DQmRkniNS7V11jSQ8wtp28Zmz87S9Z8BKDnhngPozJVXkw9/image.png)

![](https://steemitimages.com/DQmVDvbN1D1gTuspsyRmjYCjwRq58gCtorev2mb3v9zephb/image.png)

![](https://steemitimages.com/DQmXADbiLuvHM1bHQwxxYhQgbz2JqNR9ViGr64zXgDLXVAK/image.png)

![](https://steemitimages.com/DQmZcBj2oRp7pYbumCdVZ8xcp4kZgPS8F7DyC1igD9MLFtA/image.png)

![](https://steemitimages.com/DQmR1d7LMKBHxfkkX8vS2UuLVBvj7R1GaPiJSkPScse4utk/image.png)

![](https://steemitimages.com/DQmYTavGRu1AQ5FwU4hVh4amCzijNAFjVERZwqLsVuDW3nQ/image.png)

👍 hendrikdegrote, lafona-miner, mes, steemstem, pharesim, justtryme90, liberosist, curie, hr1, bue, meerkat, anwenbaumeister, masterofcoin, lemouth, fredrikaa, mobbs, anarchyhasnogods, trumpman, kushed, dber, bogh, steemstem-bot, ertwro, eric-boucher, howtostartablog, gra, dahbou, dna-replication, mountainwashere, nitesh9, gentleshaid, the-devil, jasimg, amavi, mystifact, mathiew, mariprofundus, trashing88, shanahs, horpey, ovij, foundation, locikll, ksolymosi, gohba.handcrafts, steemulator, mikees, himal, aboutyourbiz, cryptokrieg, rachelsmantra, markangeltrueman, blessing97, hadji, rjbauer85, ewuoso, coloringiship, sammarkjames, jmdk2000, ivan.atman, xanderslee, esaia.mystic, prechi

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