If `x=a(theta+sintheta),\ \ y=a(1+costheta)` , find `(dy)/(dx)` .

Explanation

Given that ` x=a(theta-sin theta), y=a(1+cos theta)`

Differentiating w.r.t. `theta`, we get

`frac{d x}{d theta}=a(1-cos theta), frac{d y}{d theta}=a(0-sin theta) `

`=-a sin theta`

`therefore frac{d y}{d x}=frac{frac{d y}{d theta}}{frac{d x}{d theta}}=frac{-a sin theta}{a(1-cos theta)}`

`=frac{-sin theta}{1-cos theta}=frac{-2 sin frac{theta}{2} cos frac{theta}{2}}{2 sin ^{2} frac{theta}{2}}=frac{-cos frac{theta}{2}}{sin frac{theta}{2}} `

`=-cot frac{theta}{2}`