The real part of `z = (1)/(1-cos theta + i sin theta)` is

Options

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

Answer

To find the real part of the complex number \( z = \frac{1}{1 - \cos \theta + i \sin \theta} \), we will follow these steps:

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

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

### Step 3: Simplify the denominator
Using the formula \( (a + b)(a - b) = a^2 - b^2 \), we can simplify the denominator:
\[ (1 - \cos \theta)^2 - (i \sin \theta)^2 = (1 - \cos \theta)^2 - (-1)(\sin^2 \theta) \]
This simplifies to:
\[ (1 - \cos \theta)^2 + \sin^2 \theta \]

### Step 4: Expand the denominator
Now we expand \( (1 - \cos \theta)^2 \):
\[ (1 - \cos \theta)^2 = 1 - 2\cos \theta + \cos^2 \theta \]
Thus, the denominator becomes:
\[ 1 - 2\cos \theta + \cos^2 \theta + \sin^2 \theta \]
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can simplify this to:
\[ 1 - 2\cos \theta + 1 = 2 - 2\cos \theta = 2(1 - \cos \theta) \]

### Step 5: Simplify the numerator
The numerator remains:
\[ 1 - \cos \theta - i \sin \theta \]

### Step 6: Write \( z \) in standard form
Now we can write \( z \) as:
\[ z = \frac{1 - \cos \theta - i \sin \theta}{2(1 - \cos \theta)} \]
This can be separated into real and imaginary parts:
\[ z = \frac{1 - \cos \theta}{2(1 - \cos \theta)} - i \frac{\sin \theta}{2(1 - \cos \theta)} \]
This simplifies to:
\[ z = \frac{1}{2} - i \frac{\sin \theta}{2(1 - \cos \theta)} \]

### Step 7: Identify the real part
The real part of \( z \) is:
\[ \text{Re}(z) = \frac{1}{2} \]

### Final Answer
Thus, the real part of \( z \) is:
\[ \frac{1}{2} \]

---