Chapter 7.3, Problem 1TFE

Label each of the following statements as either true or false.

There is a one-to-one correspondence between the standard form and the trigonometric form of a complex number.

To determine

Whether the statement, “There is a one-to-one correspondence between the standard form and the trigonometric form of a complex number” is true or false.

Answer to Problem 1TFE

Solution:

The statement, “There is a one-to-one correspondence between the standard form and the trigonometric form of a complex number” is false.

Explanation of Solution

Consider the statement, “The standard form and the trigonometric form of a complex number show one to one correspondence”.

A complex number in standard form $x+iy$ is written as $x+iy=r(\cos \theta +i\sin \theta )$ this is called the trigonometric form of $x+iy$ where $r=\sqrt{x^2+y^2}$ is called absolute value and $\theta$ is called the argument of $x+iy$.

Let $-2+2\sqrt{3}i=x+iy$,

Therefore,

$r=\sqrt{{\left(-2\right)}^2+{\left(2\sqrt{3}\right)}^2}=\sqrt{4+12}=\sqrt{16}=4$

Since $x=-2<0$ and $y=2\sqrt{3}>0$,

$\theta ={\tan }^{-1}\left(\frac{2\sqrt{3}}{-2}\right)+\pi ={\tan }^{-1}\left(-\sqrt{3}\right)+\pi =-\frac{\pi }{3}+\pi =\frac{2\pi }{3}$

$\Rightarrow \theta =\frac{2\pi }{3},\frac{2\pi }{3}+2\pi =\frac{8\pi }{3},\frac{2\pi }{3}-2\pi =-\frac{4\pi }{3}$

Therefore, $-2+2\sqrt{3}i$ can be written in trigonometric form as

$-2+2\sqrt{3}i=4\left(-\frac{2}{4}+\frac{2\sqrt{3}}{4}i\right)$

$=4\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)$

$=4\left(\cos \frac{8\pi }{3}+i\sin \frac{8\pi }{3}\right)$

$=4\left(\cos \left(-\frac{4\pi }{3}\right)+i\sin \left(-\frac{4\pi }{3}\right)\right)$

Since more than one trigonometric form represents the same complex numbers, then there is no one-to-one correspondence between the standard form and the trigonometric form of a complex number.

Therefore, “The standard form and the trigonometric form of a complex number do not show one to one correspondence”.

Hence, the given statement is false.