Find the modulus and amplitude of the following complex numbers and hence express them into polar form
`-sqrt3+i`

Answer

To find the modulus and amplitude of the complex number \( z = -\sqrt{3} + i \), and express it in polar form, we will follow these steps:

### Step 1: Identify the real and imaginary parts
The complex number \( z \) can be expressed as:
- Real part \( a = -\sqrt{3} \)
- Imaginary part \( b = 1 \)

### Step 2: Calculate the modulus
The modulus of a complex number \( z = a + ib \) is given by the formula:
\[ |z| = \sqrt{a^2 + b^2} \]
Substituting the values of \( a \) and \( b \):
\[ |z| = \sqrt{(-\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

### Step 3: Calculate the amplitude (argument)
The amplitude (or argument) \( \theta \) of the complex number can be found using:
\[ \tan(\theta) = \frac{b}{a} \]
Substituting the values:
\[ \tan(\theta) = \frac{1}{-\sqrt{3}} \]
This gives us \( \theta = -\frac{\pi}{6} \). However, since the complex number \( z \) is in the second quadrant (where the real part is negative and the imaginary part is positive), we need to adjust our angle:
\[ \theta = \pi - \frac{\pi}{6} = \frac{6\pi}{6} - \frac{\pi}{6} = \frac{5\pi}{6} \]

### Step 4: Express in polar form
The polar form of a complex number is given by:
\[ z = |z| \left( \cos(\theta) + i \sin(\theta) \right) \]
Substituting the modulus and amplitude:
\[ z = 2 \left( \cos\left(\frac{5\pi}{6}\right) + i \sin\left(\frac{5\pi}{6}\right) \right) \]

### Final Result
Thus, the modulus of \( z \) is \( 2 \), the amplitude is \( \frac{5\pi}{6} \), and the polar form of the complex number is:
\[ z = 2 \left( \cos\left(\frac{5\pi}{6}\right) + i \sin\left(\frac{5\pi}{6}\right) \right) \]

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