Express the complex number in modulus amplitudes form `1+i sqrt(3)`

Answer

To express the complex number \( z = 1 + i\sqrt{3} \) in modulus-amplitude form, we will follow these steps:

### Step 1: Identify the complex number
We have the complex number:
\[ z = 1 + i\sqrt{3} \]

### Step 2: Calculate the modulus \( r \)
The modulus \( r \) of a complex number \( z = a + bi \) is given by:
\[ r = \sqrt{a^2 + b^2} \]
Here, \( a = 1 \) and \( b = \sqrt{3} \). Thus,
\[ r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 3: Calculate the angle \( \theta \)
The angle \( \theta \) can be found using the formulas:
\[ \cos \theta = \frac{a}{r}, \quad \sin \theta = \frac{b}{r} \]
Substituting the values we have:
\[ \cos \theta = \frac{1}{2}, \quad \sin \theta = \frac{\sqrt{3}}{2} \]

### Step 4: Determine the angle \( \theta \)
From trigonometric values, we know:
\[ \cos 60^\circ = \frac{1}{2} \quad \text{and} \quad \sin 60^\circ = \frac{\sqrt{3}}{2} \]
Thus, \( \theta = 60^\circ \).

### Step 5: Write the modulus-amplitude form
The modulus-amplitude (or polar) form of a complex number is given by:
\[ z = r \left( \cos \theta + i \sin \theta \right) \]
Substituting the values we found:
\[ z = 2 \left( \cos 60^\circ + i \sin 60^\circ \right) \]

### Final Answer
The complex number \( 1 + i\sqrt{3} \) in modulus-amplitude form is:
\[ z = 2 \left( \cos 60^\circ + i \sin 60^\circ \right) \]

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