Question

In Problems 11–22, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees.

$2+\sqrt{3}i$.

Short answer

The polar form of $2+\sqrt{3}i$is, $\sqrt{7}\left(\cos 40.{89}^{\circ }+i\sin 40.{89}^{\circ }\right)$.

The given complex number is plotted in the complex plane.

Step-by-step solution

Step 1 Given argument is,

$2+\sqrt{3}i$.

The point corresponding to $z=2+\sqrt{3}i$has the rectangular coordinates $\left(2,\sqrt{3}\right)$.

The point is located in the first quadrant and is plotted in the graph.

Step 2 The point is plotted in the graph and it is shown below.

Now from the co-ordinates of the point $P\left(2,\sqrt{3}\right)$we know that,$x=2,y=\sqrt{3}$.

Step 3 By using these values we can find out the polar coordinates, r and θ of the point P.

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ =\sqrt{2^2+{\left(\sqrt{3}\right)}^2} \\ r=\sqrt{7} \end{gathered}$$

and,

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{y}{x}\right) \\ ={\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right) \\ =40.{89}^{\circ } \end{gathered}$$

Hence the polar co-ordinates are $\left(\sqrt{7},40.{89}^{\circ }\right)$and in polar form it can be written as,$z=\sqrt{7}\left(\cos 40.{89}^{\circ }+i\sin 40.{89}^{\circ }\right)$.