SOLUTION: Please help me convert a complex number to polar form and exponential form. 2+2(sqrt3i)

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Question 1122243: Please help me convert a complex number to polar form and exponential form.
2+2(sqrt3i)
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
a + bi = 2+%2B+2%2Asqrt%283%29%2Ai.


a = 2;  b = 2%2Asqrt%283%29


The mudulus  r = sqrt%28a%5E2+%2B+b%5E2%29 = sqrt%282%5E2+%2B+%282%2Asqrt%283%29%29%5E2%29 = sqrt%284+%2B+4%2A3%29 = sqrt%2816%29 = 4.



The argument  theta = arctan%28b%2Fa%29 = arctan%28%282%2Asqrt%283%29%29%2F2%29 = arctan%28sqrt%283%29%29 = pi%2F3.


Polar form:  a + bi = 2+%2B+2%2Asqrt%283%29%2Ai = (4,pi%2F3),   where first argument "4" is the modulus and the second argument  pi%2F3  is the polar angle.


It is the same as  the Trigonometric form of a complex number a + bi = 2+%2B+2%2Asqrt%283%29%2Ai = 4%2A%28cos%28pi%2F3%29+%2B+i%2Asin%28pi%2F3%29%29 = 4%2A%28%281%2F2%29+%2B+i%2A%28sqrt%283%29%2F2%29%29 = 4%2Acis%28pi%2F3%29.


Exponential form:  a + bi = 2+%2B+2%2Asqrt%283%29%2Ai = 4*e^(i*(pi/3)) = 4%2Ae%5E%28i%2A%28pi%2F3%29%29.

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On complex numbers, see the lessons
    - Complex numbers and arithmetical operations on them
    - Complex plane
    - Addition and subtraction of complex numbers in complex plane
    - Multiplication and division of complex numbers in complex plane
    - Raising a complex number to an integer power
    - How to take a root of a complex number
    - Solution of the quadratic equation with real coefficients on complex domain
    - How to take a square root of a complex number
    - Solution of the quadratic equation with complex coefficients on complex domain

    - Solved problems on taking roots of complex numbers
    - Solved problems on arithmetic operations on complex numbers
    - Solved problem on taking square root of complex number
    - Miscellaneous problems on complex numbers
    - Advanced problem on complex numbers
    - Solved problems on de'Moivre formula
    - A curious example of an equation in complex numbers which HAS NO a solution
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic  "Complex numbers".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.