Question 136

If $$x = \sin \theta + \theta \cos \theta$$ and $$y = \cos \theta + \theta \sin \theta$$, then $$\frac{dy}{dx}$$ at $$\theta = \frac{\pi}{2}$$ is

Solution

$$\frac{dy}{d\theta\ }=-\sin\theta\ +\sin\theta\ +\theta\ \cos\theta\ =\theta\ \cos\theta\ $$

$$\frac{dx}{d\theta\ }=\cos\theta\ +\cos\theta\ -\theta\ \sin\theta\ =2\cos\theta\ -\theta\ \sin\theta\ $$

$$\frac{dy}{dx}=\frac{dy}{d\theta\ }\times\ \frac{d\theta}{dx}=\frac{\theta\ \cos\theta}{2\cos\theta\ -\theta\ \sin\theta\ }$$

Substituting $$\theta\ =\frac{\pi}{2}$$ we get $$\frac{4}{\pi\ }$$

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