Trigonometric form#

We can also observe the complex plane in polar coordinates $(r, ϕ)$ which are related to the Cartesian coordinates $(x, y)$ through:

x = r \cos φ, y = r \sin φ

This leads to the trigonometric form of a complex number $z$ :

z = r (\cos φ + i \sin φ),

where $r$ is the modulus (or magnitude), equal to the absolute value of the complex number, a direct consequence of Pythagora’s theorem:

r = \sqrt{x^{2} + y^{2}} = | z |

and the polar angle $φ$ is defined (to a multiply $2 π$ ) by:

\tan (φ + 2 n π) = \frac{y}{x}, n \in Z, x \neq 0.

The angle $φ$ is called the argument (or phase) of a complex number $z$ and we write $Arg (z) = φ$ . A special case is the complex number $z = 0$ for which $r = 0$ and its argument is not defined.

Finding $φ$ is delicate because $\tan$ is a multivalued function. To avoid ambiguity, the simplest choice is $n = 0$ so that the interval is of length $2 π$ and $- π < arg (z) \leq π$ . The value of $Arg (z)$ with $n = 0$ is called the principal value of the argument. With this:

arg (1) = 0, arg (i) = \frac{π}{2}, arg (- 1) = π, arg (- i) = - \frac{π}{2}, etc.

THe relationship between $Arg (z)$ and $arg (z)$ is therefore:

Arg (z) = arg (z) + 2 n π, n \in Z .

Multiplying two complex numbers results in their absolute values being multiplied and the arguments being added:

\begin{array}{r} \begin{aligned} z_{1} \cdot z_{2} & = r_{1} (\cos φ_{1} + i \sin φ_{1}) \cdot r_{2} (\cos φ_{2} + i \sin φ_{2}) \\ = r_{1} r_{2} (\cos (φ_{1} + φ_{2}) + i \sin (φ_{1} + φ_{2})) \end{aligned} \end{array}

For the multiplicative inverse we have

\begin{array}{r} \begin{aligned} z^{- 1} & = \frac{1}{r (\cos φ + i \sin φ)} \cdot \frac{c o s φ - i \sin φ}{c o s φ - i \sin φ} = \frac{\cos φ - i \sin φ}{r (\cos^{2} φ + \sin^{2} φ)} \\ = \frac{1}{r} (\cos φ - i \sin φ) \end{aligned} \end{array}

Properties of $| z |$ and arg( $z$ )

Let us summarise our findings in the form of the following equalities:

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x		(x² − 1) (x − 1)
0.5		1.50000
0.9		1.90000
0.99		1.99000
0.999		1.99900
0.9999		1.99990
0.99999		1.99999
...		...

Limits (Formal Definition)

Approaching ...

Example Continued:

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Trigonometric form#

Limits (Formal Definition)

Approaching ...

Example Continued:

Trigonometric form#

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