Question

How do you write the trigonometric form into a complex number in standard form $3\left( \cos 120+i\sin 120 \right)?$

Answer 1

The trigonometric form of complex number can be written in the standard rom i.e. $z=a+bi$ which is rectangular form and the polar form is $z=r\left( \cos \theta +i\sin \theta \right)$ where $r=\left| a+bi \right|$ is the modules of $z$ and the angle $\theta$ can be calculated by formula $\theta ={{\tan }^{-1}}\left( \dfrac{b}{a} \right)$

Complete step-by-step answer:
So, here we have to write the trigonometric form into a complex number in a standard form.
In the problem given trigonometric form of complex number is $3\left( \cos 120+i\sin 120 \right)$
Here the value of $\cos 120$ is $\dfrac{-1}{2}$
And the value of $\sin 120$ is $\dfrac{\sqrt{3}}{2}$
Therefore, the trigonometric form of complex number $3\left( \cos 120+i\sin 120 \right)$ can be written as
$3\left( \cos 120+i\sin 120 \right)=3\left( \dfrac{-1}{2}+i\dfrac{\sqrt{3}}{2} \right)$
Now, multiplying the $3$ to the terms in the bracket. Therefore we have,
$3\left( \cos 120+i\sin 120 \right)=\dfrac{-3}{2}+i\dfrac{\sqrt{3}}{2}$

So, the trigonometric form into a complex number in standard form of $3\left( \cos 120+i\sin 120 \right)$ is $\left( \dfrac{-3}{2}+i\dfrac{3\sqrt{3}}{2} \right)$

Additional Information:
Here, Trigonometric polar form of complex number can be written as,
$z=r\left( \cos \theta +i\sin \theta \right)$
Where, $r=\left| z \right|$ and $\theta =$ angle $\left( z \right)$
Here, $z=a+bi,$ therefore modules of $z$ can be calculated as,
$\left| z \right|=\left| a+bi \right|$ which is equal to $'r'$
Therefore,
$r=\left| a+bi \right|$
$r=\left| \sqrt{{{a}^{2}}+{{b}^{2}}} \right|$
And the angle $\theta$ can be calculated as,
$\theta ={{\tan }^{-1}}\left( \dfrac{b}{a} \right)$
Then, we get the polar coordinates that is $\left( r;\theta \right)$ which help in deriving the trigonometric polar form i.e.,
Trigonometric form $=r\left( \cos \theta +i\sin \theta \right)$
Above is the trigonometric polar form of complex number but to convert the trigonometric polar form into a complex number of rectangular form or standard form. We have, $z=r\left( \cos \theta +i\sin \theta \right)$ then convert into standard rectangular form that is $z=a+bi$
Then, applying the below formula for the value of $'a'$ and $'b'$ that is.
$a=r\cos \theta$
$b=r\sin \theta$
Then put the value of $a$ and $b$ in $z=a+bi$ you will get the rectangular (standard) form.

Note:
Find the sine and cosine angle accurately while solving the problem. Multiply the $'r'$ value appropriately to the sine and cosine value. Here $'r'$ means the radius of the circle which corresponds to the hypotenuse of the right triangle for your angle $'\theta '$