If `x + iy = (1)/(1-cos theta + 2 i sin theta), theta ne 2n pi, n in I`, then maximum value of x is

Options

A. 1
B. 2
C. `(1)/(2)`
D. `(1)/(3)`

Answer

To find the maximum value of \( x \) given the expression \( x + iy = \frac{1}{1 - \cos \theta + 2i \sin \theta} \), we will follow these steps:

### Step 1: Express the complex number
We start with the expression:
\[ z = x + iy = \frac{1}{1 - \cos \theta + 2i \sin \theta} \]

### Step 2: Rationalize the denominator
To simplify this expression, we multiply the numerator and denominator by the conjugate of the denominator:
\[ z = \frac{1 \cdot (1 - \cos \theta - 2i \sin \theta)}{(1 - \cos \theta + 2i \sin \theta)(1 - \cos \theta - 2i \sin \theta)} \]

### Step 3: Simplify the denominator
The denominator can be simplified using the formula \( (a + b)(a - b) = a^2 - b^2 \):
\[ (1 - \cos \theta)^2 + (2 \sin \theta)^2 = (1 - \cos \theta)^2 + 4 \sin^2 \theta \]
Expanding \( (1 - \cos \theta)^2 \):
\[ 1 - 2\cos \theta + \cos^2 \theta + 4 \sin^2 \theta = 1 - 2\cos \theta + \cos^2 \theta + 4(1 - \cos^2 \theta) \]
This simplifies to:
\[ 1 - 2\cos \theta + 5 - 3\cos^2 \theta = 6 - 2\cos \theta - 3\cos^2 \theta \]

### Step 4: Write the real part \( x \)
Now, we can express \( z \):
\[ z = \frac{1 - \cos \theta - 2i \sin \theta}{6 - 2\cos \theta - 3\cos^2 \theta} \]
The real part \( x \) is given by:
\[ x = \frac{1 - \cos \theta}{6 - 2\cos \theta - 3\cos^2 \theta} \]

### Step 5: Find the maximum value of \( x \)
To maximize \( x \), we need to minimize the denominator \( 6 - 2\cos \theta - 3\cos^2 \theta \).

Let \( u = \cos \theta \). The expression becomes:
\[ f(u) = 6 - 2u - 3u^2 \]
This is a quadratic function that opens downwards. The maximum occurs at:
\[ u = -\frac{b}{2a} = -\frac{-2}{2 \cdot -3} = \frac{1}{3} \]

### Step 6: Calculate the minimum value of the denominator
Substituting \( u = \frac{1}{3} \) into the denominator:
\[ 6 - 2\left(\frac{1}{3}\right) - 3\left(\frac{1}{3}\right)^2 = 6 - \frac{2}{3} - 3 \cdot \frac{1}{9} = 6 - \frac{2}{3} - \frac{1}{3} = 6 - 1 = 5 \]

### Step 7: Maximum value of \( x \)
Thus, the maximum value of \( x \) is:
\[ x = \frac{1 - \cos \theta}{5} \]
To find the maximum value of \( 1 - \cos \theta \), we note that \( \cos \theta \) can range from -1 to 1. Therefore, the maximum value occurs when \( \cos \theta = -1 \):
\[ 1 - (-1) = 2 \]
Thus, the maximum value of \( x \) is:
\[ x = \frac{2}{5} \]

### Final Answer
The maximum value of \( x \) is \( \frac{2}{5} \).