Question

The real part of e^(3+4i)x is
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Answer 1

Step-by-step explanation:

the real part of e^(3+4i)x is 3

Answer 2

$\displaystyle \sf Real \: part \: of \: \: {e}^{(3 + 4i)x} = \bf {e}^{3x} cos \: 4x$

Given :

$\displaystyle \sf \: complex \: number \: \: {e}^{(3 + 4i)x}$

To find :

$\displaystyle \sf Real \: part \: of \: \: {e}^{(3 + 4i)x}$

Concept :

Complex Number

A complex number z = a + ib is defined as an ordered pair of Real numbers ( a, b) that satisfies the following conditions :

(i) Condition for equality :

(a, b) = (c, d) if and only if a = c, b = d

(ii) Definition of addition :

(a, b) + (c, d) = (a+c, b+ d)

(iii) Definition of multiplication :

(a, b). (c, d) = (ac-bd , ad+bc )

Of the ordered pair (a, b) the first component a is called Real part of z and the second component b is called Imaginary part of z

Solution :

Step 1 of 2 :

Write down the given complex number

Here the given complex number is

$\displaystyle \sf = {e}^{(3 + 4i)x}$

Step 2 of 2 :

$\displaystyle \sf Find\: real \: part \: of \: \: {e}^{(3 + 4i)x}$

$\displaystyle \sf {e}^{(3 + 4i)x}$

$\displaystyle \sf = {e}^{(3x + 4ix)}$

$\displaystyle \sf = {e}^{3x } \times {e}^{4ix}$

$\displaystyle \sf = {e}^{3x } \times (cos \: 4x + i \: sin \: 4x) \: \: \: \bigg[ \: \because \: {e}^{ibx} = cos \: bx + i \: sin \: bx\bigg]$

$\displaystyle \sf = ( {e}^{3x }cos \: 4x + i \: {e}^{3x }sin \: 4x)$

$\displaystyle \sf \therefore \: Real \: part \: of \: \: {e}^{(3 + 4i)x} = {e}^{3x} cos \: 4x$

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