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

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.

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.

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