If ` x =theta -sin theta ,y=1 -cos theta ,then " at " theta =(pi)/(2) ,(dy)/(dx) =`

Options

A. ` (-1)/(2)`
B. ` (1)/(2)`
C. ` -1`
D. ` 1`

Answer

To solve the problem, we need to find \(\frac{dy}{dx}\) at \(\theta = \frac{\pi}{2}\), given the equations \(x = \theta - \sin \theta\) and \(y = 1 - \cos \theta\).

### Step-by-Step Solution:

1. **Find \(\frac{dx}{d\theta}\)**:
\[ x = \theta - \sin \theta \]
To differentiate \(x\) with respect to \(\theta\):
\[ \frac{dx}{d\theta} = 1 - \cos \theta \]

2. **Find \(\frac{dy}{d\theta}\)**:
\[ y = 1 - \cos \theta \]
To differentiate \(y\) with respect to \(\theta\):
\[ \frac{dy}{d\theta} = 0 + \sin \theta = \sin \theta \]

3. **Find \(\frac{dy}{dx}\)** using the chain rule:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{\sin \theta}{1 - \cos \theta} \]

4. **Evaluate \(\frac{dy}{dx}\) at \(\theta = \frac{\pi}{2}\)**:
\[ \frac{dy}{dx} \bigg|_{\theta = \frac{\pi}{2}} = \frac{\sin\left(\frac{\pi}{2}\right)}{1 - \cos\left(\frac{\pi}{2}\right)} \]
Since \(\sin\left(\frac{\pi}{2}\right) = 1\) and \(\cos\left(\frac{\pi}{2}\right) = 0\):
\[ \frac{dy}{dx} \bigg|_{\theta = \frac{\pi}{2}} = \frac{1}{1 - 0} = 1 \]

### Final Answer:
\[ \frac{dy}{dx} = 1 \text{ at } \theta = \frac{\pi}{2} \]