The polar form of 1+i`sqrt(3)` is

Options

A. `2( cos ""(pi)/(6) + isin""(pi)/(6))`
B. `2( cos ""(pi)/(3) + isin""(pi)/(3))`
C. `2(cos ""(2pi)/(3)+isin""(2pi)/(3))`
D. `2( cos (-(pi)/(6))+ isin (-(pi)/(6)))`

Answer

To convert the complex number \( z = 1 + i\sqrt{3} \) into its polar form, we need to find the modulus \( r \) and the argument \( \theta \).

### Step 1: Find the modulus \( r \)

The modulus \( r \) of a complex number \( z = a + bi \) is given by:

\[ r = \sqrt{a^2 + b^2} \]

For our complex number \( z = 1 + i\sqrt{3} \), we have \( a = 1 \) and \( b = \sqrt{3} \).

Calculating \( r \):

\[ r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 2: Find the argument \( \theta \)

The argument \( \theta \) can be found using the formula:

\[ \tan \theta = \frac{b}{a} \]

For our complex number:

\[ \tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3} \]

Now, we need to find \( \theta \) such that \( \tan \theta = \sqrt{3} \). The angle \( \theta \) that satisfies this in the first quadrant is:

\[ \theta = \frac{\pi}{3} \]

### Step 3: Write the polar form

The polar form of a complex number is given by:

\[ z = r(\cos \theta + i \sin \theta) \]

Substituting the values of \( r \) and \( \theta \):

\[ z = 2\left(\cos\frac{\pi}{3} + i \sin\frac{\pi}{3}\right) \]

### Final Answer

Thus, the polar form of \( 1 + i\sqrt{3} \) is:

\[ z = 2\left(\cos\frac{\pi}{3} + i \sin\frac{\pi}{3}\right) \]

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