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Question
Find an angle θ whose rate of increase twice is twice the rate of decrease of its cosine ?
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Solution
\[\text { Let x } = \cos\theta\]
\[\text { Differentiating both sides with respect to t, we get }\]
\[\frac{d x}{d t} = \frac{d \left( \cos\theta \right)}{d t}\]
\[ = - \sin\theta\frac{d \theta}{d t}\]
\[\text { But it is given that } \frac{d \theta}{d t} = - 2\frac{d x}{d t}\]
\[ \Rightarrow \frac{d x}{d t} = - \sin\theta\left( - 2\frac{d x}{d t} \right)\]
\[ \Rightarrow \sin\theta = \frac{1}{2}\]
\[ \Rightarrow \theta = \frac{\pi}{6}\]
\[\text { Hence }, \theta = \frac{\pi}{6} .\]
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