What is the argument of `(1-sintheta)+icos theta` ?
`(i=sqrt(-1))`

Options

A. `(pi)/(2)-(theta)/(2)`
B. `(pi)/(2)+(theta)/(2)`
C. `(pi)/(4)-(theta)/(2)`
D. `(pi)/(4)+(theta)/(2)`

Explanation

To find the argument of the complex number \( Z = (1 - \sin \theta) + i \cos \theta \), we can follow these steps:

### Step 1: Identify the real and imaginary parts
The complex number can be expressed as:
- Real part: \( \text{Re}(Z) = 1 - \sin \theta \)
- Imaginary part: \( \text{Im}(Z) = \cos \theta \)

### Step 2: Use the formula for the argument
The argument \( \arg(Z) \) of a complex number is given by:
\[ \arg(Z) = \tan^{-1}\left(\frac{\text{Im}(Z)}{\text{Re}(Z)}\right) \]
Substituting the real and imaginary parts, we have:
\[ \arg(Z) = \tan^{-1}\left(\frac{\cos \theta}{1 - \sin \theta}\right) \]

### Step 3: Simplify the expression
To simplify \( \frac{\cos \theta}{1 - \sin \theta} \), we can multiply the numerator and denominator by \( 1 + \sin \theta \):
\[ \frac{\cos \theta (1 + \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)} = \frac{\cos \theta (1 + \sin \theta)}{1 - \sin^2 \theta} = \frac{\cos \theta (1 + \sin \theta)}{\cos^2 \theta} \]
This simplifies to:
\[ \frac{1 + \sin \theta}{\cos \theta} \]

### Step 4: Find the argument
Now we can express the argument as:
\[ \arg(Z) = \tan^{-1}\left(\frac{1 + \sin \theta}{\cos \theta}\right) \]
Using the identity \( \tan^{-1}(x) = \tan^{-1}(y) \) where \( y = \tan \left(\frac{\pi}{4} + \frac{\theta}{2}\right) \), we can write:
\[ \arg(Z) = \frac{\pi}{4} + \frac{\theta}{2} \]

### Final Result
Thus, the argument of the complex number \( Z \) is:
\[ \arg(Z) = \frac{\pi}{4} + \frac{\theta}{2} \]

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