Question

How do you find the trigonometric form of a complex number?

Answer 1

Iota (“i”) is known as the complex number whose value is minus root one, it was found by the mathematician to deal the negative sign under root, previously when this was not defined then if negative sign comes under root then there was no solution for that, but after this research complex terms can now easily be solved and tackle.

Formula used:
$z\, = \,r(\cos \theta \, + \,i\sin \theta )\,where\,\,r\, = \,\left| z \right|\,and\,\theta \, = \,angle(z)$

Complete step by step answer:
The trigonometric form of a complex number $z\, = \,a\, + \,bi$ is $z\, = \,r(\cos \theta \, + \,i\sin \theta )$ where $r\, = \,\left| {a\, + \,bi} \right|$ is the modulus of $z$ and $\tan \theta \, = \,b.a$. Let the complex number be $z\, = \,(x\, + \,iy)$.Polar form is $(r,\theta )$
$r\, = \,\left| {\sqrt {{x^{2\,}}\, + \,{y^2}} } \right|$
$\Rightarrow \theta \, = \,\arctan \left( {\dfrac{y}{x}} \right)$
Trigonometric form used is $r(\cos \theta \, + \,i\sin \theta )$
If we let go one real-valued and the other leg equal to the Pythagorean theorem changes to ${a^2}\, - \,{b^2}\, = \,{c^2}$ .
Except for 0, any complex number can be represented in the trigonometric form or in polar coordinates:
$z\, = \,r(\cos \theta \, + \,i\sin \theta )\,where\,\,r\, = \,\left| {a\, + \,bi} \right|$ is the modulus or the absolute value of $z$ , which is easy to find.

Additional information:
Dealing with the complex equation you have to be careful only when you are dealing in higher degree equations because there the value of “iota” is given as for higher degree terms and accordingly the question needs to be solved.

Note: After the development of iota, research leads with the formulas associated and the properties like summation, subtraction, multiplication and division for the complex numbers. Graphs for complex numbers are also designed and the area under which the graph is drawn contains complex numbers only, but the relation between complex and real numbers can be drawn.