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The subtangent is a segment STST on the X-axis from point SS, an abscissa of some point PP on a curve, to point TT, intersection of a tangent to a curve at point PP with the X-axis.
Knowing parametric expressions for x=f(t)x=f(t) and y=g(t)y=g(t), for any parameter tt we know abscissa and ordinate of a point P(t)P(t) on a curve.
In this case we will use ordinate y=a(1-cos t)y=a(1−cost) and will determine tan(Psi)tan(Ψ). That will be sufficient to find STST.
To calculate tan(Psi)=dy/dxtan(Ψ)=dydx we will use the property of parametric curve:
dy/dx = (dy/dt)/(dx/dt)dydx=dydtdxdt
Using this approach,
tan(Psi) = dy/dx = [a(sin t)]/[a(1+cos t)] = sin t/(1+cos t)tan(Ψ)=dydx=a(sint)a(1+cost)=sint1+cost
Now we can calculate STST:
ST = PS*tan(Psi) = y*dy/dx = a(1-cos t)*sin t/(1+cos t)ST=PS⋅tan(Ψ)=y⋅dydx=a(1−cost)⋅sint1+cost
This does not resemble any of the answers provided.
However, if x=a(t-sin t)x=a(t−sint) then dx/dt = 1-cos tdxdt=1−cost and answer asin tasint would be correct.
