Question

Let \(s=\cos \theta .\) Evaluate \(d s / d t\) when \(\theta=3 \pi / 2\) and \(d \theta / d t=5\)

Short answer

The value of \( ds/dt \) when \( \theta = 3\pi/2 \) and \( d\theta/dt = 5 \) is 5.

Step-by-step solution

Derive \( s \) with respect to \( \theta \)

Differentiate the function \( s = \cos \theta \) with respect to \( \theta \). The derivative of \( \cos \theta \) is \( -\sin \theta \). Therefore, \( ds/d\theta = -\sin \theta \).

Substitute \(\theta = 3\pi/2 \) into \( ds/d\theta \)

Next, evaluate \( ds/d\theta \) at the specific value \( \theta = 3\pi/2 \). Substituting this value in, we get \( ds/d\theta = -\sin(3\pi/2) \). Because \( \sin(3\pi/2) = -1 \), \( ds/d\theta = 1 \).

Apply the chain rule

Using the chain rule, \( ds/dt = ds/d\theta * d\theta/dt \). We are given that \( d\theta/dt = 5 \), and from Step 2 we found that \( ds/d\theta = 1 \). Substituting these values in yields \( ds/dt = 1 * 5 = 5 \).