If `z_(1)=-1,z_(2)=i` then find Arg `((z_(1))/(z_(2)))`

Answer

To find the argument of the complex number \(\frac{z_1}{z_2}\) where \(z_1 = -1\) and \(z_2 = i\), we can follow these steps:

### Step 1: Identify the complex numbers
We have:
- \(z_1 = -1\)
- \(z_2 = i\)

### Step 2: Use the formula for the argument of the quotient of two complex numbers
The argument of the quotient of two complex numbers is given by:
\[ \text{Arg}\left(\frac{z_1}{z_2}\right) = \text{Arg}(z_1) - \text{Arg}(z_2) \]

### Step 3: Find the argument of \(z_1\)
For \(z_1 = -1\):
- The point \(-1\) lies on the negative real axis.
- The argument of \(-1\) is \(\pi\) (or \(180^\circ\)).

Thus,
\[ \text{Arg}(z_1) = \pi \]

### Step 4: Find the argument of \(z_2\)
For \(z_2 = i\):
- The point \(i\) lies on the positive imaginary axis.
- The argument of \(i\) is \(\frac{\pi}{2}\) (or \(90^\circ\)).

Thus,
\[ \text{Arg}(z_2) = \frac{\pi}{2} \]

### Step 5: Substitute the arguments into the formula
Now we can substitute the values we found into the formula:
\[ \text{Arg}\left(\frac{z_1}{z_2}\right) = \text{Arg}(z_1) - \text{Arg}(z_2) = \pi - \frac{\pi}{2} \]

### Step 6: Simplify the expression
Now we simplify:
\[ \pi - \frac{\pi}{2} = \frac{2\pi}{2} - \frac{\pi}{2} = \frac{\pi}{2} \]

### Final Answer
Thus, the argument of \(\frac{z_1}{z_2}\) is:
\[ \text{Arg}\left(\frac{z_1}{z_2}\right) = \frac{\pi}{2} \]

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