Question

Find `(d^2y)/(dx^2)` of the following : x = a(θ – sin θ), y = a(1 – cos θ)

Answer 1

x = a(θ – sin θ), y = a(1 – cos θ)

Differentiating x and y w.r.t. θ, we get

`"dx"/"dθ" = a"d"/"dθ"(θ - sin θ)`

= a(1 – cos θ)                                       ...(1)

and

`"dy"/"dθ" = a"d"/"dθ"(1 - cos θ)`

= a[0 – (– sin θ)]

= a sin θ

∴ `"dy"/"dx" = (("dy"/"dθ"))/(("dx"/"dθ")`

= `"a sin θ"/"a(1 - cos θ)"`

= `(2sin(θ/2).cos(θ/2))/(2sin^2(θ/2)) = cot(θ/2)`

and

`(d^2y)/(dx^2) = "d"/"dx"[cot(θ/2)]`

= `"d"/"dx"[cot(θ/2)].("d"θ/2)/"dx"]`

= `-"cosec"^2(θ/2)."d"/"dθ"(θ/2) xx (1)/(("dx"/"dθ")`

= `-"cosec"^2(θ/2) xx (1)/(2) xx (1)/(a(1 - cosθ)`  ...[by (1)]

= `-(1)/(2a)"cosec"^2(θ/2) xx (1)/(2sin^2(θ/2)`

=`-(1)/(4a)."cosec"^4(θ/2)`.