If x = a cos theta and y = a sin theta,then f | vedclass

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(A) Given $x = a \cos 	heta$ and $y = a \sin 	heta$. First,find $\frac{dx}{d	heta} = -a \sin 	heta$ and $\frac{dy}{d	heta} = a \cos 	heta$. Then

Vedclass · Accepted Answer

$-\frac{1}{a} \operatorname{cosec}^3 	heta$

Vedclass · Answer

(A) Given $x = a \cos 	heta$ and $y = a \sin 	heta$. First,find $\frac{dx}{d	heta} = -a \sin 	heta$ and $\frac{dy}{d	heta} = a \cos 	heta$. Then,$\frac{dy}{dx} = \frac{dy/d	heta}{dx/d	heta} = \frac{a \cos 	heta}{-a \sin 	heta} = -\cot 	heta$. Now,differentiate $\frac{dy}{dx}$ with respect to $x$: $\frac{d^2 y}{dx^2} = \frac{d}{dx}(-\cot 	heta) = \frac{d}{d	heta}(-\cot 	heta) \cdot \frac{d	heta}{dx}$. Since $\frac{d}{d	heta}(-\cot 	heta) = \operatorname{cosec}^2 	heta$ and $\frac{d	heta}{dx} = \frac{1}{dx/d	heta} = \frac{1}{-a \sin 	heta}$. Therefore,$\frac{d^2 y}{dx^2} = \operatorname{cosec}^2 	heta \cdot \left( -\frac{1}{a \sin 	heta} ight) = -\frac{1}{a} \operatorname{cosec}^3 	heta$.

If $x = a \cos \theta$ and $y = a \sin \theta$,then $\frac{d^2 y}{d x^2} =$ . . . . . . . (where $a \neq 0$ and $\theta \neq k \pi, k \in Z$)

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