If x= and y= , prove that dy dx =1 3 at t=2 3 - Vidyadip

Question

If $\text{x}=\cot\text{t and y}=\sin\text{t},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{1}{\sqrt{3}}\text{ at t}=\frac{2\pi}{3}$

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Answer

We have, $\text{x}=\cos\text{t}$ and $\text{y}=\sin\text{t}$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=\frac{\text{d}}{\text{dt}}(\cos\text{t}) $ and $\frac{\text{dy}}{\text{dt}}=\frac{\text{d}}{\text{dt}}(\sin\text{t}) $
$\Rightarrow\frac{\text{dx}}{\text{dt}}=-\sin\text{t}$ and $\frac{\text{dy}}{\text{dt}}=\cos\text{t}$
$\therefore\frac{\frac{\text{dy}}{\text{dt}}}{\frac{\text{dx}}{\text{dt}}}=\frac{\cos\text{t}}{-\sin\text{t}}=-\cot{\text{t}}$
Now, $\big(\frac{\text{dy}}{\text{dx}}\big)_{\text{t}=\frac{2\pi}{3}}=-\cot\big(\frac{2\pi}{3}\big)=\frac{1}{\sqrt{3}}$

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