If ` x = a( sin theta - theta cos theta) and y = a ( cos theta + theta sin theta) " find " (dy)/(dx) at theta = pi/4`

Answer

To find \(\frac{dy}{dx}\) at \(\theta = \frac{\pi}{4}\) given the equations \(x = a(\sin \theta - \theta \cos \theta)\) and \(y = a(\cos \theta + \theta \sin \theta)\), we will follow these steps:

### Step 1: Differentiate \(x\) with respect to \(\theta\)

Given:
\[ x = a(\sin \theta - \theta \cos \theta) \]

Using the product rule and chain rule:
\[ \frac{dx}{d\theta} = a\left(\cos \theta - \left(\cos \theta + \theta(-\sin \theta)\right)\right) \]
\[ = a\left(\cos \theta - \cos \theta + \theta \sin \theta\right) \]
\[ = a\theta \sin \theta \]

### Step 2: Differentiate \(y\) with respect to \(\theta\)

Given:
\[ y = a(\cos \theta + \theta \sin \theta) \]

Using the product rule:
\[ \frac{dy}{d\theta} = a\left(-\sin \theta + \left(\sin \theta + \theta \cos \theta\right)\right) \]
\[ = a\left(-\sin \theta + \sin \theta + \theta \cos \theta\right) \]
\[ = a\theta \cos \theta \]

### Step 3: Find \(\frac{dy}{dx}\)

Using the chain rule:
\[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{a\theta \cos \theta}{a\theta \sin \theta} \]
\[ = \frac{\cos \theta}{\sin \theta} = \cot \theta \]

### Step 4: Evaluate \(\frac{dy}{dx}\) at \(\theta = \frac{\pi}{4}\)

Now substituting \(\theta = \frac{\pi}{4}\):
\[ \frac{dy}{dx} = \cot\left(\frac{\pi}{4}\right) = 1 \]

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

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