Find the equations of the tangent and the normal to the curve `x=theta+sintheta` , `y=1+costheta` at `theta=pi//2` at indicated points.

Explanation

`x=theta+sin theta text { and } y=1+cos theta `

`frac{d x}{d theta}=1+cos theta text { and } frac{d y}{d theta}=-sin theta `

`therefore frac{d y}{d x}=frac{frac{d y}{d theta}}{frac{d x}{d theta}}=frac{-sin theta}{1+cos theta}`

Slope of tangent `m=(frac{d y}{d x})_{theta=frac{pi}{2}}=frac{-sin frac{pi}{2}}{1+cos frac{pi}{2}}=frac{-1}{1+0}=-1` Now, `(x_{1}, y_{1})=(frac{pi}{2}+sin frac{pi}{2}, 1+cos frac{pi}{2})=(frac{pi}{2}+1,1)`
Equation of tangent is ,


`y-y_{1}=m(x-x_{1}) `

`Rightarrow y-1=-1(x-frac{pi}{2}-1) `

`Rightarrow 2 y-2=-2 x+pi+2 `

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