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Question
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
Solution
\[x = 3\sin t - \sin3t \text{ and y } = 3\cos t - \cos3t\]
\[ \Rightarrow \frac{dx}{dt} = 3\cos t - 3\cos3t \text{ and } \frac{dy}{dt} = - 3\sin t + 3\sin3t\]
\[ \therefore \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{- 3\sin t + 3\sin3t}{3\cos t - 3\cos3t}\]
\[ \Rightarrow \left( \frac{dy}{dx} \right)_{t = \frac{\pi}{3}} = \frac{- 3\sin\frac{\pi}{3} + 3sin\pi}{3\cos\frac{\pi}{3} - 3cos\pi}\]
\[ = \frac{- 3 \times \frac{\sqrt{3}}{2} + 0}{3 \times \frac{1}{2} + 3}\]
\[ = \frac{\frac{- 3\sqrt{3}}{2}}{\frac{9}{2}}\]
\[ = - \frac{1}{\sqrt{3}}\]
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