The argument of `((1-isqrt(3)))/((1+isqrt(3)))` is

Options

A. A) `60^(@)`
B. B) `120^(@)`
C. C) `210^(@)`
D. D) `240^(@)`

Answer

To find the argument of the complex number \(\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\), we can follow these steps:

### Step 1: Write the complex number in standard form
The given complex number is \(\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\).

### Step 2: Multiply numerator and denominator by the conjugate of the denominator
To simplify, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(1 - i\sqrt{3}\):
\[ \frac{(1 - i\sqrt{3})(1 - i\sqrt{3})}{(1 + i\sqrt{3})(1 - i\sqrt{3})} \]

### Step 3: Simplify the numerator
Calculating the numerator:
\[ (1 - i\sqrt{3})(1 - i\sqrt{3}) = 1 - 2i\sqrt{3} + (i\sqrt{3})^2 = 1 - 2i\sqrt{3} - 3 = -2 - 2i\sqrt{3} \]

### Step 4: Simplify the denominator
Calculating the denominator:
\[ (1 + i\sqrt{3})(1 - i\sqrt{3}) = 1 - (i\sqrt{3})^2 = 1 - (-3) = 1 + 3 = 4 \]

### Step 5: Combine the results
Now we can write the expression as:
\[ \frac{-2 - 2i\sqrt{3}}{4} = -\frac{1}{2} - \frac{\sqrt{3}}{2}i \]

### Step 6: Identify the real and imaginary parts
From \(-\frac{1}{2} - \frac{\sqrt{3}}{2}i\), we identify:
- Real part \(x = -\frac{1}{2}\)
- Imaginary part \(y = -\frac{\sqrt{3}}{2}\)

### Step 7: Determine the quadrant
Since both the real part and the imaginary part are negative, the complex number lies in the third quadrant.

### Step 8: Find the reference angle
The reference angle can be found using:
\[ \cos \theta = -\frac{1}{2} \quad \text{and} \quad \sin \theta = -\frac{\sqrt{3}}{2} \]
The angles corresponding to these values are \(60^\circ\) (for cosine) and \(60^\circ\) (for sine).

### Step 9: Calculate the argument
Since the complex number is in the third quadrant, we add \(180^\circ\) to the reference angle:
\[ \text{Argument} = 180^\circ + 60^\circ = 240^\circ \]

### Final Answer
Thus, the argument of the complex number \(\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\) is \(240^\circ\).

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