The sum of the lengths of the subtangent and | vedclass

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(C) Given $	heta = \frac{\pi}{3}$ and the cycloid equations $x = a(	heta - \sin 	heta)$,$y = a(1 - \cos 	heta)$.First,find the derivatives with re

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$\frac{2 a}{\sqrt{3}}$

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(C) Given $	heta = \frac{\pi}{3}$ and the cycloid equations $x = a(	heta - \sin 	heta)$,$y = a(1 - \cos 	heta)$.First,find the derivatives with respect to $	heta$:$\frac{dx}{d	heta} = a(1 - \cos 	heta)$ and $\frac{dy}{d	heta} = a \sin 	heta$.Then,the slope of the tangent is $\frac{dy}{dx} = \frac{dy/d	heta}{dx/d	heta} = \frac{a \sin 	heta}{a(1 - \cos 	heta)} = \frac{\sin 	heta}{1 - \cos 	heta}$.At $	heta = \frac{\pi}{3}$,the slope $m = \frac{\sin(\pi/3)}{1 - \cos(\pi/3)} = \frac{\sqrt{3}/2}{1 - 1/2} = \sqrt{3}$.The value of $y$ at $	heta = \frac{\pi}{3}$ is $y = a(1 - \cos(\pi/3)) = a(1 - 1/2) = \frac{a}{2}$.Length of subtangent $= \left| \frac{y}{m} ight| = \frac{a/2}{\sqrt{3}} = \frac{a}{2\sqrt{3}}$.Length of subnormal $= |y \cdot m| = \frac{a}{2} \cdot \sqrt{3} = \frac{a\sqrt{3}}{2}$.Sum of the lengths $= \frac{a}{2\sqrt{3}} + \frac{a\sqrt{3}}{2} = \frac{a + 3a}{2\sqrt{3}} = \frac{4a}{2\sqrt{3}} = \frac{2a}{\sqrt{3}}$.

The sum of the lengths of the subtangent and the subnormal drawn at $\theta = \frac{\pi}{3}$ on the cycloid $x = a(\theta - \sin \theta)$,$y = a(1 - \cos \theta)$ is

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