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

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(A) Given $x = a \cos 	heta$ and $y = a \sin 	heta$.First, find the derivatives with respect to $	heta$:$\frac{dx}{d	heta} = -a \sin 	heta$ and $

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$-\frac{1}{a} \csc^3 	heta$

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(A) Given $x = a \cos 	heta$ and $y = a \sin 	heta$.First, find the derivatives with respect to $	heta$:$\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 $x$ using the chain rule:$\frac{d^2 y}{dx^2} = \frac{d}{dx}(-\cot 	heta) = \frac{d}{d	heta}(-\cot 	heta) \cdot \frac{d	heta}{dx}$.Since $\frac{d}{d	heta}(-\cot 	heta) = \csc^2 	heta$ and $\frac{d	heta}{dx} = \frac{1}{dx/d	heta} = \frac{1}{-a \sin 	heta}$, we have:$\frac{d^2 y}{dx^2} = (\csc^2 	heta) \cdot \left( \frac{1}{-a \sin 	heta} ight) = -\frac{1}{a} \csc^3 	heta$.

If $x = a \cos \theta$ and $y = a \sin \theta$, then $\frac{d^2 y}{d x^2} = \rule{1cm}{0.15mm} \, (a \neq 0; \theta \neq k\pi, k \in Z)$.

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