The complex number in trigonometric form is [tex]z = 2(cos\frac{\pi }{2} +isin\frac{\pi }{2} )[/tex]
What is a complex number?
A complex number is characterized by an ordered pair of numbers, the real and imaginary parts. It can be represented in multiple ways such as Standard form(rectangular form), Polar form(trigonometric form), Exponential form.
For the given situation,
The complex number is 2i
The trigonometric form of a complex number is
[tex]z =a+bi= r(cos\theta+isin\theta)[/tex]
where [tex]r=|z|=\sqrt{(a^2+b^2) }[/tex], [tex]a = rcos\theta[/tex] and [tex]b = rsin\theta[/tex], and [tex]\theta = tan^{-1} (\frac{b}{a} )[/tex]
Here, [tex]z = a + bi = 0 + 2i[/tex]
a = 0, b = 2
Then, [tex]r=|z|=\sqrt{(0^2+2^2) }[/tex]
⇒ [tex]r=|z|=\sqrt{(0+4) }[/tex]
⇒ [tex]r=|z|=\sqrt{4}[/tex]
⇒ [tex]r=|z|=2[/tex]
Now, [tex]\theta = tan^{-1} (\frac{2}{0} )[/tex]
⇒ [tex]\theta = tan^{-1} ( \infty )[/tex]
⇒ [tex]\theta =\frac{\pi }{2}[/tex]
Thus, [tex]z = r(cos\theta+isin\theta)[/tex] becomes
⇒ [tex]z = 2(cos\frac{\pi }{2} +isin\frac{\pi }{2} )[/tex]
Hence we can conclude that the complex number in trigonometric form is [tex]z = 2(cos\frac{\pi }{2} +isin\frac{\pi }{2} )[/tex]
Learn more about complex number here
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