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Question

(A) Given equations are $x = \cos 	heta - \cos 2	heta$ and $y = \sin 	heta - \sin 2	heta$. Differentiating $x$ with respect to $	heta$: $\frac{dx

Vedclass · Accepted Answer

$\frac{\cos 	heta - 2 \cos 2	heta}{2 \sin 2	heta - \sin 	heta}$

Vedclass · Answer

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

If $x$ and $y$ are connected parametrically by the equations,without eliminating the parameter,find $\frac{dy}{dx}$ for $x = \cos \theta - \cos 2\theta$ and $y = \sin \theta - \sin 2\theta$.

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