An angle \( \theta,0\lt\theta\lt\displaystyle\frac{\pi}{2} \), which increases twice as fast as its sine, is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{\pi}{3} \)
“θ increases twice as fast as sin θ” means, with respect to time t,
\( \displaystyle \frac{d\theta}{dt}=2\,\frac{d}{dt}(\sin\theta) \)
\( \displaystyle \frac{d\theta}{dt}=2\cos\theta\,\frac{d\theta}{dt} \)
Since θ is changing, \( \frac{d\theta}{dt}\ne 0 \), and it cancels:
\( 1=2\cos\theta \)
\( \displaystyle \cos\theta=\frac{1}{2} \)
\( \displaystyle \theta=\frac{\pi}{3}\qquad\left(0\lt\theta\lt\frac{\pi}{2}\right) \)
NCERT Exemplar:
Application of Derivatives
