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

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Last updated date: 23rd May 2024
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Answer
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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 $2$.
$
  x = 2\left( {\dfrac{{1 + i\sqrt 3 }}{2}} \right) = 2\left( {\dfrac{1}{2}} \right) + 2\left( {\dfrac{{i\sqrt 3 }}{2}} \right) \\
  x = 2\cos \dfrac{\pi }{3} + i2\sin \left[ {\dfrac{\pi }{3}} \right] \\
 $
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,
$x = 2{e^{i\dfrac{\pi }{3}}}$
Hence, $2{e^{i\dfrac{\pi }{3}}}$ 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.