Question

Find an angle θ whose rate of increase twice is twice the rate of decrease of its cosine ?

Answer 1

\[\text { Let x } = \cos\theta\]

\[\text { Differentiating both sides with respect to t, we get }\]

\[\frac{d x}{d t} = \frac{d \left( \cos\theta \right)}{d t}\]

\[ = - \sin\theta\frac{d \theta}{d t}\]

\[\text { But it is given that } \frac{d \theta}{d t} = - 2\frac{d x}{d t}\]

\[ \Rightarrow \frac{d x}{d t} = - \sin\theta\left( - 2\frac{d x}{d t} \right)\]

\[ \Rightarrow \sin\theta = \frac{1}{2}\]

\[ \Rightarrow \theta = \frac{\pi}{6}\]

\[\text { Hence }, \theta = \frac{\pi}{6} .\]