Find frac{dy}{dx},if x = a(theta + sin theta) | vedclass

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(A) Given parametric equations are $x = a(	heta + \sin 	heta)$ and $y = a(1 - \cos 	heta)$.Differentiating $x$ with respect to $	heta$:$\frac{dx}{

Vedclass · Accepted Answer

$	an \frac{	heta}{2}$

Vedclass · Answer

(A) Given parametric equations are $x = a(	heta + \sin 	heta)$ and $y = a(1 - \cos 	heta)$.Differentiating $x$ with respect to $	heta$:$\frac{dx}{d	heta} = a(1 + \cos 	heta)$.Differentiating $y$ with respect to $	heta$:$\frac{dy}{d	heta} = a(0 - (-\sin 	heta)) = a \sin 	heta$.Using the chain rule for parametric differentiation:$\frac{dy}{dx} = \frac{dy/d	heta}{dx/d	heta} = \frac{a \sin 	heta}{a(1 + \cos 	heta)}$.Using trigonometric identities $\sin 	heta = 2 \sin \frac{	heta}{2} \cos \frac{	heta}{2}$ and $1 + \cos 	heta = 2 \cos^2 \frac{	heta}{2}$:$\frac{dy}{dx} = \frac{2 \sin \frac{	heta}{2} \cos \frac{	heta}{2}}{2 \cos^2 \frac{	heta}{2}} = \frac{\sin \frac{	heta}{2}}{\cos \frac{	heta}{2}} = 	an \frac{	heta}{2}$.

Find $\frac{dy}{dx}$,if $x = a(\theta + \sin \theta)$ and $y = a(1 - \cos \theta)$.

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