A cycloid is para,etrized by the equations x = r( t - sint) y = r(1 - cost)?

Question

a) find an equation of the tangent to the cycloid at point where t = 60 p=3.14.................

b)at what point is the tangent horizontal?at what points is it vertical?

Answer 1

x=r(t-sint) and y= r(1-cost) are the parametric equations of the cycloid where r is the radius of the rolling circle and t is the angle through which the radius of the circle has turned from its starting point. The angle t must be measured in radians.

For simplicity let r =1 so that we have the unit circle.
with t=60 = pi/3 we get
x = pi/3-sin(pi/3) = pi/3 -(sqrt 3)/2
y = 1-cos (pi/3) = 1- 1/2 = 1/2

dy/dx= sint/(1-cost)
=[(sqrt 3)/2]/(1/2) = 3^0.5 (at the point where t= 60= pi/3)

From this , you can find the slope at any point define by the parameter t.

b) The tangent is horizontal when t= pi and is perpendicular at the cusp which occurs at 2pi and multiples thereof. The maximum value is 2r (or 2 for the unit circle). The length of the base of the arch is 2pi *r or just 2pi for the unit circle.