Question

Find an angle θ which increases twice as fast as its cosine ?

Answer 1

\[\text { Let x  }= \cos\theta\]
\[\text { Differentiating both sides with respect to t, we get }\]
\[\frac{d x}{d t} = \frac{d \left( \cos\theta \right)}{d t}\]
\[ = - \sin\theta\frac{d \theta}{d t}\]
\[\text { But it is given that } \frac{d \theta}{d t} = 2\frac{d x}{d t}\]
\[ \Rightarrow \frac{d x}{d t} = - \sin\theta\left( 2\frac{d x}{d t} \right)\]
\[ \Rightarrow \sin\theta = - \frac{1}{2}\]
\[ \Rightarrow \theta = \pi + \frac{\pi}{6} = \frac{7\pi}{6}\]
\[\text { Hence,} \theta = \frac{7\pi}{6} .\]