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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 21 (page 591)

In Problems $11 - 22$ , plot each complex number in the complex plane and write it in polar form. Express the argument in degrees.
$- 2 + 3 i$ .

Short Answer

Expert verified

The polar form of $- 2 + 3 i$ is, $\sqrt{13} (\cos 123 . 7^{鈭�} + i \sin 123 . 7^{鈭�})$ .

The given complex number is plotted in the complex plane.

Step by step solution

01

Step 1 Given argument is,

$- 2 + 3 i$ .

The rectangular co-ordinates of the point $(- 2 + 3 i)$ is, $(- 2, 3)$ .

02

Step 2 The point is plotted in the complex plane.

From the graph it is clear that the point is located in the first quadrant.

03

Step 3 We know that,

$|z| = r r = \sqrt{x^{2} + y^{2}} r = \sqrt{{(- 2)}^{2} + 3^{2}} r = \sqrt{13}$

Use $x = r \cos 胃, y = r \sin 胃$ to find the value of $胃$ .

$\sin 胃 = \frac{y}{r} = \frac{3}{\sqrt{13}} \cos 胃 = \frac{x}{r} = \frac{- 2}{\sqrt{13}}$

Since $\sin 胃$ is $\frac{3}{\sqrt{13}}$ and $\cos 胃 = \frac{- 2}{\sqrt{13}}$ ,the value of $胃$ is, $123 . 7^{鈭�}$ .

04

Step 4 The polar form of z=x+iy is, rcos θ+i sin θ.

The polar form of $- 2 + 3 i$ is, $\sqrt{13} (\cos 123 . 7^{鈭�} + i \sin 123 . 7^{鈭�})$ .

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