If `sqrt(1- sin A)= sin "" A/2 - cos "" A/2 ` could lie in quadrant

Options

A. first
B. second
C. third
D. fourth

Answer

To solve the equation \( \sqrt{1 - \sin A} = \sin \frac{A}{2} - \cos \frac{A}{2} \) and determine in which quadrants it can lie, we can follow these steps:

### Step 1: Simplify the Left Side
We start with the left side of the equation:
\[ \sqrt{1 - \sin A} \]
Using the Pythagorean identity, we know that:
\[ 1 - \sin A = \cos^2 \frac{A}{2} \]
Thus, we can rewrite the left side:
\[ \sqrt{1 - \sin A} = \sqrt{\cos^2 \frac{A}{2}} = |\cos \frac{A}{2}| \]

### Step 2: Analyze the Right Side
Now, let's analyze the right side:
\[ \sin \frac{A}{2} - \cos \frac{A}{2} \]
This expression can be rewritten as:
\[ \sin \frac{A}{2} - \cos \frac{A}{2} = \frac{\sqrt{2}}{2}(\sqrt{2} \sin \frac{A}{2} - \sqrt{2} \cos \frac{A}{2}) = \sqrt{2} \left(\sin \frac{A}{2} - \cos \frac{A}{2}\right) \]

### Step 3: Set Up the Equation
Now we equate both sides:
\[ |\cos \frac{A}{2}| = \sin \frac{A}{2} - \cos \frac{A}{2} \]

### Step 4: Solve the Inequality
To solve for when this holds true, we need to consider the signs of \( \sin \frac{A}{2} \) and \( \cos \frac{A}{2} \).

1. **Case 1**: If \( \cos \frac{A}{2} \geq 0 \):
\[ \cos \frac{A}{2} = \sin \frac{A}{2} - \cos \frac{A}{2} \]
This simplifies to:
\[ 2\cos \frac{A}{2} = \sin \frac{A}{2} \]
or
\[ \tan \frac{A}{2} = 2 \]

2. **Case 2**: If \( \cos \frac{A}{2} < 0 \):
\[ -\cos \frac{A}{2} = \sin \frac{A}{2} - \cos \frac{A}{2} \]
This simplifies to:
\[ 0 = \sin \frac{A}{2} \]
which gives \( \frac{A}{2} = n\pi \) where \( n \) is an integer.

### Step 5: Determine Quadrants
From the equation \( \tan \frac{A}{2} = 2 \), we find the angles for \( \frac{A}{2} \):
- The solutions for \( \tan \frac{A}{2} = 2 \) occur in the first and third quadrants.

Thus, \( \frac{A}{2} \) can be in the first quadrant (where \( A \) is in the first quadrant) or in the third quadrant (where \( A \) is in the third quadrant).

Therefore, \( A \) can lie in the following quadrants:
- **2nd Quadrant**: \( \frac{A}{2} \) is in the 1st quadrant.
- **3rd Quadrant**: \( \frac{A}{2} \) is in the 3rd quadrant.
- **4th Quadrant**: \( \frac{A}{2} \) can also be in the 4th quadrant.

### Conclusion
The values of \( A \) can lie in the 2nd, 3rd, and 4th quadrants.