Question

If $y=1-\cos \theta ,x=1-\sin \theta ,$ then $\frac{dy}{dx}$ at $\theta =\frac{\pi }{4}$ is

• A. $-1$
• B. $1$
• C. $\frac{1}{2}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

Answer: (A)

$-1$

$y=1-\cos \theta$

$\frac{dy}{d\theta }=0-(-\sin \theta )=\sin \theta$

$x=1-\sin \theta$

$\frac{dx}{d\theta }=0-\left(\cos \theta \right)=-\cos \theta$

Therefore,

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

$\frac{dy}{dx}=\frac{\sin \theta }{-\cos \theta }=-\tan \theta$

Since, $\theta =\frac{\pi }{4}$

Therefore,

$-\tan \frac{\pi }{4}=-1$

$Hence,optionAisthecorrectanswer.$