Question

If $x=a(\cos t+\log \tan \frac{t}{2})andy=a\sin t,$ then find $\frac{d^{2}y}{d{x}^2}$ at $t=\frac{n}{3}$

Answer 1

$x=a(cost+\log \tan \frac{t}{2}),y=asint$

Calculating $\frac{dy}{dt}$

$y=asint$

$\frac{dy}{dt}=\frac{d}{dt}\left(asint\right)$

$\frac{dy}{dt}=acost$ ------- ( 1 )

Now calculating $\frac{dx}{dt}$

$x=a(\cos t+\log tan\frac{t}{2})$

$\frac{dx}{dt}=\frac{d}{dt}a(\cos t+\log \tan \frac{t}{2})$

$\frac{dx}{dt}=a(-sint+\frac{1}{\tan \frac{t}{2}}.\frac{d\left(\tan \frac{t}{2}\right)}{dt})$

$\frac{dx}{dt}=a(-\sin t+\frac{1}{\tan \frac{t}{2}}.({\sec }^2\frac{t}{2}).\frac{d\left(\frac{t}{2}\right)}{dt})$

$\frac{dx}{dt}=a(-\sin t+\frac{1}{\tan \frac{t}{2}}.se{c}^2(\frac{t}{2}).\frac{1}{2})$

$\frac{dx}{dt}=a(-\sin t+\cot \frac{t}{2}.\frac{1}{{\cos }^2\frac{t}{2}}.\frac{1}{2})$

$\frac{dx}{dt}=a(-\sin t+\frac{\cos \frac{t}{2}}{\sin \frac{t}{2}}.\frac{1}{co{s}^2\frac{t}{2}}.\frac{1}{2})$

$\frac{dx}{dt}=a(-sint+\frac{1}{\sin \frac{t}{2}\cos \frac{t}{2}}.\frac{1}{2})$

$\frac{dx}{dt}=a(-\sin t+\frac{1}{2\sin \frac{t}{2}\cos \frac{t}{2}})$

$\frac{dx}{dt}=a(-\sin t+\frac{1}{\sin t})$

$\frac{dx}{dt}=a\left(\frac{-si{n}^{2}t+1}{\sin t}\right)$

$\frac{dx}{dt}=a\left(\frac{1-{\sin }^{2}t}{\sin t}\right)$

$\frac{dx}{dt}=a\left(\frac{{\cos }^{2}t}{\sin t}\right)$ ----( 2 )

From ( 1 ) and ( 2 )

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

$\frac{dy}{dx}=\frac{a\cos t}{\frac{a{\cos }^{2}t}{\sin t}}$

$\frac{dy}{dx}=a\cos t\times \frac{\sin t}{a{\cos }^{2}t}$

$\frac{dy}{dx}=\frac{a\cos t.\sin t}{acost.\cos t}$

$\Rightarrow$ $\frac{dy}{dx}=\frac{\sin t}{\cos t}$

$\Rightarrow$ $\frac{dy}{dx}=\tan t$

$\therefore$ $\frac{d^{2}y}{d{x}^2}=\frac{d}{dt}(\frac{dt}{dx}\times \frac{dy}{dx})$

$\Rightarrow$ $\frac{d^{2}y}{d{x}^2}=\frac{d}{dt}(\frac{\sin t}{a{\cos }^{2}t}\times \tan t)$

$\Rightarrow$ $\frac{d^{2}y}{d{x}^2}=\frac{1}{a}\frac{d}{dt}\left(\frac{{\sin }^{2}t}{{\cos }^{3}t}\right)$

$\frac{d^{2}y}{d{x}^2}=\frac{co{s}^{3}t.2sint.cost-si{n}^{2}t.3{\cos }^{2}t.(-sint)}{a(co{s}^{3}t{)}^2}$

$\frac{d^{2}y}{d{x}^2}=\frac{cost(sin2t.co{s}^{2}t+3si{n}^{2}t)}{aco{s}^{6}t}$

$\frac{d^{2}y}{d{x}^2}=\frac{(sin2t.co{s}^{2}t+3si{n}^{2}t)}{aco{s}^{5}t}$

Now substitude $t=\frac{n}{3}$

$\frac{d^{2}y}{d{x}^2}=\frac{(sin\frac{2n}{3}.co{s}^2\frac{n}{3}+3si{n}^2\frac{n}{3})}{aco{s}^5\frac{n}{3}}$