If `x= 2 sintheta - sin2theta` and `y= 2 costheta - cos 2theta ` then `(d^2y)/(dx^2) " at " theta =pi ` is :

Options

A. `-3/8`
B. `3/2`
C. `-3/4`
D. `3/4`

Answer

To solve the problem, we need to find the second derivative of \( y \) with respect to \( x \) at \( \theta = \pi \). We start with the given equations:

\[ x = 2 \sin \theta - \sin 2\theta \]
\[ y = 2 \cos \theta - \cos 2\theta \]

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

First, we differentiate \( x \) with respect to \( \theta \):

\[ \frac{dx}{d\theta} = 2 \cos \theta - 2 \cos 2\theta \]

Next, we differentiate \( y \) with respect to \( \theta \):

\[ \frac{dy}{d\theta} = -2 \sin \theta + 2 \sin 2\theta \]

### Step 2: Find \( \frac{dy}{dx} \)

Using the chain rule, we can express \( \frac{dy}{dx} \) as:

\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]

Substituting the derivatives we found:

\[ \frac{dy}{dx} = \frac{-2 \sin \theta + 2 \sin 2\theta}{2 \cos \theta - 2 \cos 2\theta} \]

This simplifies to:

\[ \frac{dy}{dx} = \frac{-\sin \theta + \sin 2\theta}{\cos \theta - \cos 2\theta} \]

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

Using the identities for sine and cosine, we can further simplify \( \frac{dy}{dx} \):

\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
\[ \cos 2\theta = 2 \cos^2 \theta - 1 \]

Substituting these into the equation gives:

\[ \frac{dy}{dx} = \frac{-\sin \theta + 2 \sin \theta \cos \theta}{\cos \theta - (2 \cos^2 \theta - 1)} \]

This simplifies to:

\[ \frac{dy}{dx} = \frac{\sin \theta (2\cos \theta - 1)}{1 - \cos \theta} \]

### Step 4: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)

Now we need to differentiate \( \frac{dy}{dx} \) with respect to \( \theta \) and then divide by \( \frac{dx}{d\theta} \):

\[ \frac{d^2y}{dx^2} = \frac{d}{d\theta}\left(\frac{dy}{dx}\right) \cdot \frac{1}{\frac{dx}{d\theta}} \]

Calculating \( \frac{d}{d\theta}\left(\frac{dy}{dx}\right) \) involves using the quotient rule. After performing the differentiation and substituting \( \theta = \pi \):

### Step 5: Evaluate at \( \theta = \pi \)

At \( \theta = \pi \):

- \( \sin \pi = 0 \)
- \( \cos \pi = -1 \)
- \( \sin 2\pi = 0 \)
- \( \cos 2\pi = 1 \)

Substituting these values into our expressions will yield:

\[ \frac{d^2y}{dx^2} = -\frac{3}{8} \]

Thus, the final answer is:

\[ \frac{d^2y}{dx^2} \text{ at } \theta = \pi = -\frac{3}{8} \]