Question

If $x=2(\theta +\sin \theta )$ and $y=2(1-\cos \theta )$, then value of $\frac{dy}{dx}$ is

• A. $\tan \frac{\theta }{2}$
• B. $\cot \frac{\theta }{2}$
• C. $\sin \frac{\theta }{2}$
• D. $\cos \frac{\theta }{2}$

Answer 1

The correct option is A $\tan \frac{\theta }{2}$

$x=a(\theta +\sin \theta )y=a(1-\cos \theta )$

$\frac{dx}{dy}=a(1+\cos \theta )dy(d\theta =a\sin \theta )$

$\frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }=\frac{\sin \theta }{1+\cos \theta }=\frac{2\cos \theta /2\sin \theta /2}{1+2\cos 2\theta /2-1}$

$\frac{dy}{dx}=\frac{2\sin \theta /2\cos \theta /2}{2{\cos }^2\theta /2}=\frac{\sin \theta /2}{\cos \theta /2}$

$\frac{dy}{dx}=\tan \theta /2$