The complex number \( z \) is given by \( =\sqrt{3}+i \), what is the polar representation of \( z \) ? Select one: a. \( \quad \sqrt{2}\left(\cos \left(\frac{\pi}{6}\right)+i \sin \left(\frac{\pi}{6}\right)\right) \) b. \( \quad \sqrt{2}\left(\cos \left(\frac{\pi}{6}\right)-i \sin \left(\frac{\pi}{6}\right)\right) \) c. \( \quad 2\left(\cos \left(\frac{\pi}{6}\right)-i \sin \left(\frac{\pi}{6}\right)\right) \) d. \( \quad 2\left(\cos \left(\frac{\pi}{6}\right)+i \sin \left(\frac{\pi}{6}\right)\right) \)

To find the polar representation of the complex number \( z = \sqrt{3} + i \), we need to determine its magnitude \( r \) and angle \( \theta \). 1. The magnitude \( r \) is given by: \[ r = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2. \] 2. The angle \( \theta \) is obtained using the arctangent function: \[ \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}. \] Now, expressing \( z \) in polar form: \[ z = r \left( \cos(\theta) + i \sin(\theta) \right) = 2 \left( \cos\left(\frac{\pi}{6}\right) + i \sin\left(\frac{\pi}{6}\right) \right). \] Therefore, the correct answer is: d. \( \quad 2\left(\cos \left(\frac{\pi}{6}\right)+i \sin \left(\frac{\pi}{6}\right)\right) \). Now, let's add some fun facts! In ancient history, the concept of complex numbers was largely ignored until the 16th century when it became useful in solving cubic equations. Mathematicians like Gerolamo Cardano began incorporating imaginary numbers into their calculations, paving the way for modern mathematics! Want to impress your friends? Try explaining that complex numbers have real-world applications in electrical engineering! Engineers use them to analyze AC circuits by representing voltage and current as complex numbers, making it easier to understand phase differences and oscillations! Who knew math could be so cool?