To convert the given complex number [tex]\( 2 \sqrt{3} + 2i \)[/tex] into its polar representation, let's follow these steps:
### Step 1: Identify the real and imaginary parts
The given complex number is [tex]\( 2 \sqrt{3} + 2i \)[/tex].
- The real part ([tex]\( a \)[/tex]) is [tex]\( 2 \sqrt{3} \)[/tex].
- The imaginary part ([tex]\( b \)[/tex]) is 2.
### Step 2: Calculate the magnitude [tex]\( r \)[/tex]
The magnitude [tex]\( r \)[/tex] (also known as the modulus) of a complex number can be found using the formula:
[tex]\[ r = \sqrt{a^2 + b^2} \][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ r = \sqrt{(2 \sqrt{3})^2 + 2^2} \][/tex]
[tex]\[ r = \sqrt{(2 \sqrt{3})^2 + 4} \][/tex]
[tex]\[ r = \sqrt{4 \cdot 3 + 4} \][/tex]
[tex]\[ r = \sqrt{12 + 4} \][/tex]
[tex]\[ r = \sqrt{16} \][/tex]
[tex]\[ r = 4 \][/tex]
### Step 3: Calculate the angle [tex]\( \theta \)[/tex]
The angle [tex]\( \theta \)[/tex] (also known as the argument) can be found using the formula:
[tex]\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ \theta = \tan^{-1}\left(\frac{2}{2 \sqrt{3}}\right) \][/tex]
[tex]\[ \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \][/tex]
The arctangent of [tex]\( \frac{1}{\sqrt{3}} \)[/tex] is [tex]\( 30^\circ \)[/tex] or [tex]\( \frac{\pi}{6} \)[/tex] radians.
### Step 4: Write the polar form
The polar form of a complex number is given by:
[tex]\[ r \left( \cos \theta + i \sin \theta \right) \][/tex]
Substituting the values we have calculated:
[tex]\[ 4 \left( \cos 30^\circ + i \sin 30^\circ \right) \][/tex]
### Conclusion
The polar form of [tex]\( 2 \sqrt{3} + 2i \)[/tex] is [tex]\( 4 \left( \cos 30^\circ + i \sin 30^\circ \right) \)[/tex].
Hence, the correct answer is:
C. [tex]\( 4 \left( \cos 30^\circ + i \sin 30^\circ \right) \)[/tex]
