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If \( x=a(\theta+\sin \theta), y=a(1-\cos \theta) \), prove that \( \frac{d y}{d x}=\tan \left(\frac{\theta}{2}\right) \) |
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Answer» \(Given : x=a(θ+sinθ) \) and \(y=a(1-cosθ)\) Differentiating w.r.t. t we get \({dx \over dt} =a({dθ \over dt}) + cosθ({dθ \over dt })\)••••• (1) And \({dy \over dt } = asin({dθ \over dt} )\) •••••(2) \({dy \over dx} = {{dy \over dt} \over {dx \over dt} } = {a{dθ \over dt}(sinθ) \over a{dθ \over dt} (1+cosθ) }\) ••••from (1) and (2) \( = { sinθ \over 1+cosθ }\) \(= {2sin({θ \over 2}) cos({θ \over 2 }) \over 2 cos²({θ \over 2}) }\) \(= {sin({θ \over 2}) \over cos( {θ \over 2}) }\) \(= tan({θ \over 2})\) |
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