If `z = sqrt(20i - 21) + sqrt(20i + 21)`, than one of the possible value of |arg(z)| equals

Options

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

Answer

To solve the problem, we need to find the argument of the complex number \( z = \sqrt{20i - 21} + \sqrt{20i + 21} \).

### Step-by-Step Solution:

1. **Define \( z \)**:
\[ z = \sqrt{20i - 21} + \sqrt{20i + 21} \]

2. **Square \( z \)**:
To eliminate the square roots, we square both sides:
\[ z^2 = \left( \sqrt{20i - 21} + \sqrt{20i + 21} \right)^2 \]
Using the formula \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ z^2 = (20i - 21) + (20i + 21) + 2\sqrt{(20i - 21)(20i + 21)} \]

3. **Simplify the expression**:
\[ z^2 = 20i - 21 + 20i + 21 + 2\sqrt{(20i)^2 - 21^2} \]
\[ z^2 = 40i + 2\sqrt{400i^2 - 441} \]
Since \( i^2 = -1 \):
\[ z^2 = 40i + 2\sqrt{-400 - 441} = 40i + 2\sqrt{-841} \]
\[ z^2 = 40i + 2(29i) = 40i + 58i = 98i \]

4. **Find the argument of \( z^2 \)**:
The argument of a purely imaginary number \( yi \) is given by:
\[ \arg(yi) = \frac{\pi}{2} \text{ if } y > 0 \]
Thus:
\[ \arg(z^2) = \arg(98i) = \frac{\pi}{2} \]

5. **Relate the argument of \( z^2 \) to \( z \)**:
We know that:
\[ \arg(z^2) = 2 \arg(z) \]
Therefore:
\[ 2 \arg(z) = \frac{\pi}{2} \]
Dividing both sides by 2:
\[ \arg(z) = \frac{\pi}{4} \]

6. **Find the absolute value of the argument**:
Since we are asked for \( | \arg(z) | \):
\[ | \arg(z) | = \frac{\pi}{4} \]

### Conclusion:
One of the possible values of \( | \arg(z) | \) equals \( \frac{\pi}{4} \).