Find the equation


Find the equation of the tangent to the curve $x=\theta+\sin \theta, y=1+\cos \theta$ at $\theta=\pi / 4$.


finding slope of the tangent by differentiating $x$ and $y$ with respect to theta

$\frac{\mathrm{dx}}{\mathrm{d} \theta}=1+\cos \theta$

$\frac{\mathrm{dy}}{\mathrm{d} \theta}=-\sin \theta$

Dividing both the above equations

$\frac{d y}{d x}=-\frac{\sin \theta}{1+\cos \theta}$

$m$ at theta $(\pi / 4)=-1+\frac{1}{\sqrt{2}}$

equation of tangent is given by $y-y_{1}=m(\operatorname{tangent})\left(x-x_{1}\right)$


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