Question

If x = a sin θ and y = b cos θ, then $\frac{d^{2}y}{d{x}^2}$ = ______________________.

Answer 1

Given, $x=a\sin \theta$ and $y=b\cos \theta$.

$x=a\sin \theta$

Differentiating both sides with respect to θ, we get

$\frac{dx}{d\theta }=a\cos \theta$

$y=b\cos \theta$

Differentiating both sides with respect to θ, we get

$\frac{dy}{d\theta }=-b\sin \theta$

$\therefore \frac{dy}{dx}=\frac{{\displaystyle \frac{dy}{d\theta }}}{{\displaystyle \frac{dx}{d\theta }}}$

$\Rightarrow \frac{dy}{dx}=\frac{{\displaystyle -b\sin \theta }}{{\displaystyle a\cos \theta }}$

$\Rightarrow \frac{dy}{dx}=-\frac{{\displaystyle b\tan \theta }}{{\displaystyle a}}$

Differentiating both sides with respect to x, we get

$\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(-\frac{b\tan \theta }{a}\right)$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=-\frac{b}{a}{\sec }^2\theta \frac{d\theta }{dx}$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=-\frac{b}{a}{\sec }^2\theta \times \frac{1}{a\cos \theta }\left(\frac{dx}{d\theta }=a\cos \theta \Rightarrow \frac{d\theta }{dx}=\frac{1}{a\cos \theta }\right)$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=-\frac{b}{a^2}{\sec }^3\theta$


If x = a sin θ and y = b cos θ, then $\frac{d^{2}y}{d{x}^2}$ = $\underline{-\frac{b}{a^2}{\sec }^3\theta }$.