In this video I go further into the derivation of the cycloid curve's parametric equations and this time go over the case where the angle which the circle rotates is between π/2 and π. The only difference in the derivation for this case, as compared with my earlier derivation for the case where the angle is between 0 and π/2 is that we are instead having to subtract the angle from π radians. But we use trig identities or the definition of trigonometric functions to show that we can write sine and cosine in exactly the same way as in my last video. This is an interesting video on showing how trig functions can be written and manipulated in different ways while still getting the same result, so make sure to watch this video!
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Example:
Derive the parametric equations for a cycloid for the case π/2 < θ < π.
Solution:
In my last video (https://youtu.be/XmHpIZqeRKk I derived the parametric equations for the cycloid for the case 0 < θ < π/2.
Assume the circle of radius r rolls a distance rθ (recall that this corresponds to the arc length and rotation θ):
