Question

Express the complex number $1+i\sqrt{3}$ in modulus amplitude form.

Answer 1

Hint: Divide and multiply a number so that the complex number can be expressed in terms of sine & cosine of angles.

Lets say, $x=1+i\sqrt{3}$
Multiply and divide the RHS of the above equation with $2$.
$$\begin{gathered} x=2\left({\displaystyle \frac{1+i\sqrt{3}}{2}}\right)=2\left({\displaystyle \frac{1}{2}}\right)+2\left({\displaystyle \frac{i\sqrt{3}}{2}}\right) \\ x=2\cos {\displaystyle \frac{\pi }{3}}+i2\sin \left[{\displaystyle \frac{\pi }{3}}\right] \end{gathered}$$
This above equation can be written in exponential form
As we know $\cos \theta +i\sin \theta =e^{i\theta }$
Doing the same in the equation obtained we get,
$x=2e^{i{\displaystyle \frac{\pi }{3}}}$
Hence, $2e^{i{\displaystyle \frac{\pi }{3}}}$ in modulus amplitude form.

Note :- In these types of questions we have to obtain the given equation in the form of $\cos \theta +i\sin \theta =e^{i\theta }$ to convert it into modulus amplitude form. We should also be aware of trigonometric values needed to convert the equation in general form.