Question

Find `"dy"/"dx"` of the following function:

x = a(θ – sin θ), y = a(1 – cos θ)

Answer 1

x = a(θ – sin θ)   y = a(1 – cos θ)

`"dx"/("d"theta)` = a(1 - cosθ), `"dy"/("d"theta)` = a(0 - (- sin θ)) = a sin θ

`"dy"/"dx" = ("dy"/("d"theta))/("dx"/("d"theta))`

`= ("a" (2sin  theta/2 cos  theta/2))/(2 sin^2  theta/2)`

`= (cos  theta/2)/(sin  theta/2)`

`= cot  theta/2`

∵ sin θ = 2 sin `theta/2` cos `theta/2`

1 - cos θ = 2 `sin^2  theta/2`