Find d y-d x if y-121-cos t- x-10t-sin t-x-t-2-

Given, y=12(1 cos t),x=10(t sin t) Differentiating both equations w.r.t. t, we get ∴ dy/dt=12(0+sin t) and dx/dt=10(1 cos t) dy/dx=dy/dt/dx/dt=dy/dt×dt/dx=12 si ...

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Byju's Answer

Standard XII

Mathematics

Special Integrals - 1

Find d y/d x ...

Question

$F i n d \frac{d y}{d x} i f y = 12 (1 - c o s t), x = 10 (t - s i n t), - \frac{π}{x} < t < \frac{π}{2} .$

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Solution

Given, $y = 12 (1 - c o s t), x = 10 (t - s i n t)$

Differentiating both equations w.r.t. t, we get

$∴ \frac{d y}{d t} = 12 (0 + s i n t) a n d \frac{d x}{d t} = 10 (1 - c o s t) \frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}} = \frac{d y}{d t} \times \frac{d t}{d x} = \frac{12 s i n t}{10 (1 - c o s t)} = \frac{12 \times 2 s i n (t / 2) c o s (t / 2)}{10 {2 s i n^{2} (t / 2)}} = \frac{6}{5} c o t (\frac{t}{2}) (∴ s i n t = 2 s i n \frac{t}{2} c o s \frac{t}{x} a n d c o s t = 1 - 2 s i n^{2} \frac{t}{2})$

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Find dydx if y=12(1−cos t),x=10(t−sin t),−πx<t<π2.

Given, y=12(1−cos t),x=10(t−sin t) Differentiating both equations w.r.t. t, we get ∴ dydt=12(0+sin t) and dxdt=10(1−cos t)dydx=dydtdxdt=dydt×dtdx=12 sin t10(1−cos t)=12×2 sin (t/2)cos (t/2)10{2 sin2(t/2)}=65cot(t2) (∴ sin t=2 sin t2 cos tx and cos t=1−2sin2t2)

$F i n d \frac{d y}{d x} i f y = 12 (1 - c o s t), x = 10 (t - s i n t), - \frac{π}{x} < t < \frac{π}{2} .$