If `x=sinthetacos2theta,y=costhetasin2theta," then "((dy)/(dx))" at "theta=(pi)/(4)` is

Options

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

Answer

To find \(\frac{dy}{dx}\) at \(\theta = \frac{\pi}{4}\) given \(x = \sin \theta \cos 2\theta\) and \(y = \cos \theta \sin 2\theta\), we will use the chain rule.

### Step-by-Step Solution:

1. **Differentiate \(x\) with respect to \(\theta\)**:
\[ x = \sin \theta \cos 2\theta \]
Using the product rule:
\[ \frac{dx}{d\theta} = \frac{d}{d\theta}(\sin \theta) \cdot \cos 2\theta + \sin \theta \cdot \frac{d}{d\theta}(\cos 2\theta) \]
The derivative of \(\sin \theta\) is \(\cos \theta\) and the derivative of \(\cos 2\theta\) is \(-2\sin 2\theta\). Thus,
\[ \frac{dx}{d\theta} = \cos \theta \cos 2\theta - 2\sin \theta \sin 2\theta \]

2. **Differentiate \(y\) with respect to \(\theta\)**:
\[ y = \cos \theta \sin 2\theta \]
Using the product rule:
\[ \frac{dy}{d\theta} = \frac{d}{d\theta}(\cos \theta) \cdot \sin 2\theta + \cos \theta \cdot \frac{d}{d\theta}(\sin 2\theta) \]
The derivative of \(\cos \theta\) is \(-\sin \theta\) and the derivative of \(\sin 2\theta\) is \(2\cos 2\theta\). Thus,
\[ \frac{dy}{d\theta} = -\sin \theta \sin 2\theta + 2\cos \theta \cos 2\theta \]

3. **Evaluate \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) at \(\theta = \frac{\pi}{4}\)**:
- Calculate \(\frac{dx}{d\theta}\):
\[ \frac{dx}{d\theta} = \cos\left(\frac{\pi}{4}\right) \cos\left(\frac{\pi}{2}\right) - 2\sin\left(\frac{\pi}{4}\right) \sin\left(\frac{\pi}{2}\right) \]
Since \(\cos\left(\frac{\pi}{2}\right) = 0\) and \(\sin\left(\frac{\pi}{2}\right) = 1\):
\[ \frac{dx}{d\theta} = 0 - 2 \cdot \frac{1}{\sqrt{2}} \cdot 1 = -\sqrt{2} \]

- Calculate \(\frac{dy}{d\theta}\):
\[ \frac{dy}{d\theta} = -\sin\left(\frac{\pi}{4}\right) \sin\left(\frac{\pi}{2}\right) + 2\cos\left(\frac{\pi}{4}\right) \cos\left(\frac{\pi}{2}\right) \]
Since \(\cos\left(\frac{\pi}{2}\right) = 0\):
\[ \frac{dy}{d\theta} = -\frac{1}{\sqrt{2}} \cdot 1 + 0 = -\frac{1}{\sqrt{2}} \]

4. **Calculate \(\frac{dy}{dx}\)**:
Using the chain rule:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{-\frac{1}{\sqrt{2}}}{-\sqrt{2}} = \frac{1}{2} \]

### Final Answer:
\[ \frac{dy}{dx} \text{ at } \theta = \frac{\pi}{4} \text{ is } \frac{1}{2} \]