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Q. 20

Expert-verified
Found in: Page 591

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# In Problems 11–22, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees.$2+\sqrt{3}i$.

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

The given complex number is plotted in the complex plane.

See the 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.

$r=\sqrt{{x}^{2}+{y}^{2}}\phantom{\rule{0ex}{0ex}}=\sqrt{{2}^{2}+{\left(\sqrt{3}\right)}^{2}}\phantom{\rule{0ex}{0ex}}r=\sqrt{7}$

and,

$\theta ={\mathrm{tan}}^{-1}\left(\frac{y}{x}\right)\phantom{\rule{0ex}{0ex}}={\mathrm{tan}}^{-1}\left(\frac{\sqrt{3}}{2}\right)\phantom{\rule{0ex}{0ex}}=40.{89}^{\circ }$

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