Question
Find an angle $\theta$
Which increases twice as fast as its cosine.
Which increases twice as fast as its cosine.
✓
Answer
Let $\text{x}=\cos\theta$
Differentiating both sides with respect to t, we get
$\frac{\text{dx}}{\text{dt}}=\frac{\text{d}(\cos\theta)}{\text{dt}}$
$=-\sin\theta\frac{\text{d}\theta}{\text{dt}}$
But it is given that $\frac{\text{d}\theta}{\text{dt}}=2\frac{\text{dx}}{\text{dt}}$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=-\sin\theta\Big(2\frac{\text{dx}}{\text{dt}}\Big)$
$\Rightarrow\sin\theta=-\frac{1}{2}$
$\Rightarrow\theta=\pi+\frac{\pi}{6}=\frac{7\pi}{6}$
Hence, $\theta=\frac{7\pi}{6}.$
Differentiating both sides with respect to t, we get
$\frac{\text{dx}}{\text{dt}}=\frac{\text{d}(\cos\theta)}{\text{dt}}$
$=-\sin\theta\frac{\text{d}\theta}{\text{dt}}$
But it is given that $\frac{\text{d}\theta}{\text{dt}}=2\frac{\text{dx}}{\text{dt}}$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=-\sin\theta\Big(2\frac{\text{dx}}{\text{dt}}\Big)$
$\Rightarrow\sin\theta=-\frac{1}{2}$
$\Rightarrow\theta=\pi+\frac{\pi}{6}=\frac{7\pi}{6}$
Hence, $\theta=\frac{7\pi}{6}.$
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