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Question: If \(x = a(\cos\theta + \theta\sin\theta)\), \(y = a(\sin\theta - \theta\cos\theta),\frac{dy}{dx} =\...

If x=a(cos⁡θ+θsin⁡θ)x = a(\cos\theta + \theta\sin\theta), y=a(sin⁡θ−θcos⁡θ),dydx=y = a(\sin\theta - \theta\cos\theta),\frac{dy}{dx} =

A

cos⁡θ\cos\theta

B

tan⁡θ\tan\theta

C

sec⁡θ\sec\theta

D

cosecθ

Answer

tan⁡θ\tan\theta

Explanation

Solution

dydx=dy/dθdx/dθ\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}=a[cos⁡θ−θ(−sin⁡θ)−cos⁡θ]a[−sin⁡θ+θcos⁡θ+sin⁡θ]=θsin⁡θθcos⁡θ=tan⁡θ\frac{a\lbrack\cos\theta - \theta( - \sin\theta) - \cos\theta\rbrack}{a\lbrack - \sin\theta + \theta\cos\theta + \sin\theta\rbrack} = \frac{\theta\sin\theta}{\theta\cos\theta} = \tan\theta.