Express $z_{1}=6+2 \sqrt{3} i$ in polar form. $\newline$ Express your answer in exact terms, using degrees, where your angle is between $0^{\circ}$ and $360^{\circ}$ , inclusive. $\newline$ $z_{1}=$

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Q. Express

z_{1}=6+2 \sqrt{3} i

in polar form.

\newline

Express your answer in exact terms, using degrees, where your angle is between

0^{\circ}

and

360^{\circ}

, inclusive.

\newline

z_{1}=

Identify rectangular coordinates: Identify the rectangular coordinates of the complex number. $\newline$ The complex number $z_1 = 6 + 2\sqrt{3}i$ has a real part (x-coordinate) of $6$ and an imaginary part (y-coordinate) of $2\sqrt{3}$ .
Calculate magnitude: Calculate the magnitude (r) of the complex number. $\newline$ The magnitude is given by $r = \sqrt{x^2 + y^2}$ . $\newline$ For $z_1$ , $r = \sqrt{6^2 + (2\sqrt{3})^2}$ . $\newline$ $r = \sqrt{36 + 12}$ . $\newline$ $r = \sqrt{48}$ . $\newline$ $r = 4\sqrt{3}$ .
Calculate argument in radians: Calculate the argument (θ) of the complex number in radians. $\newline$ The argument is given by $\theta = \arctan\left(\frac{y}{x}\right)$ . $\newline$ For $z_1$ , $\theta = \arctan\left(\frac{2\sqrt{3}}{6}\right)$ . $\newline$ $\theta = \arctan\left(\frac{\sqrt{3}}{3}\right)$ . $\newline$ Since $\arctan\left(\frac{\sqrt{3}}{3}\right)$ corresponds to $\frac{\pi}{6}$ radians, we have $\theta = \frac{\pi}{6}$ .
Convert argument to degrees: Convert the argument from radians to degrees. $\newline$ $\theta$ in degrees is given by $\theta_{deg} = \theta \times \frac{180}{\pi}$ . $\newline$ For $z_1$ , $\theta_{deg} = \frac{\pi}{6} \times \frac{180}{\pi}$ . $\newline$ $\theta_{deg} = 30^{\circ}$ .
Express in polar form: Express the complex number in polar form. $\newline$ The polar form of a complex number is given by $r(\cos(\theta) + i\sin(\theta))$ . $\newline$ For $z_1$ , the polar form is $4\sqrt{3}(\cos(30^{\circ}) + i\sin(30^{\circ}))$ .