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Find \( \large\frac{dy}{dx} \), if \( y = 12 ( 1 - \cos t ) , x = 10 ( t - \sin t ) , -\large\frac{\pi}{2}\normalsize < t <\large \frac{\pi}{2} \)

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  • $\large\frac{dy}{dx}=\frac{dy}{dt}$$\times\large\frac{dt}{dx}$
Step 1:
Differentiating with respect to $t$
$\large\frac{dy}{dt}$$=12(\sin t)$
$x=10(t-\sin t)$
Differentiating with respect to $t$
$\large\frac{dx}{dt}$$=10(1-\cos t)$
Step 2:
$\large\frac{dy}{dx}=\frac{dy}{dt}$$\times\large\frac{dt}{dx}$
$\quad\;=12\sin t\times \large\frac{1}{10(1-\cos t)}$
$\quad\;=\large\frac{6.2\sin\Large\frac{t}{2}\cos\Large\frac{t}{2}}{10.2\sin^2\Large\frac{t}{2}}$
$\quad\;=\large\frac{3}{5}\frac{\cos\Large\frac{t}{2}}{\sin\Large\frac{t}{2}}$
$\quad\;=\large\frac{3}{5}$$\cot\large\frac{t}{2}$
answered May 14, 2013 by sreemathi.v
 

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