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

Last updated date: 27th Mar 2023

•

Total views: 309.9k

•

Views today: 5.86k

Answer

Verified

309.9k+ views

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.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE