The amplitude of `(-2)/(1+isqrt(3))` is

Options

A. `(pi)/(4)`
B. `(pi)/(3)`
C. `(2pi)/(3)`
D. `-(2pi)/(3)`

Answer

To find the amplitude of the complex number \(-\frac{2}{1 + i\sqrt{3}}\), we will follow these steps:

### Step 1: Rewrite the Complex Number
Let \( z = -\frac{2}{1 + i\sqrt{3}} \).

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

### Step 3: Simplify the Denominator
Using the formula \( (a + b)(a - b) = a^2 - b^2 \):
\[ (1 + i\sqrt{3})(1 - i\sqrt{3}) = 1^2 - (i\sqrt{3})^2 = 1 - (-3) = 1 + 3 = 4 \]

### Step 4: Simplify the Numerator
Now, simplify the numerator:
\[ -2(1 - i\sqrt{3}) = -2 + 2i\sqrt{3} \]

### Step 5: Combine the Results
Now we can write \( z \):
\[ z = \frac{-2 + 2i\sqrt{3}}{4} = -\frac{1}{2} + \frac{i\sqrt{3}}{2} \]

### Step 6: Identify Real and Imaginary Parts
From the expression \( z = -\frac{1}{2} + \frac{i\sqrt{3}}{2} \), we identify:
- Real part \( a = -\frac{1}{2} \)
- Imaginary part \( b = \frac{\sqrt{3}}{2} \)

### Step 7: Calculate the Amplitude
The amplitude (or argument) of a complex number is given by:
\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]
Substituting the values:
\[ \theta = \tan^{-1}\left(\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\right) = \tan^{-1}(-\sqrt{3}) \]

### Step 8: Determine the Angle
The angle \(\tan^{-1}(-\sqrt{3})\) corresponds to an angle in the second quadrant since the real part is negative and the imaginary part is positive. The reference angle for \(\tan^{-1}(\sqrt{3})\) is \(\frac{\pi}{3}\), thus:
\[ \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

### Final Answer
Thus, the amplitude of \( z \) is:
\[ \text{Amplitude} = \frac{2\pi}{3} \]

---