If-x-is-real-show-that-2-j-e-1-j3-x-2-j-e-1-j3-x-is-also-real-

Question Number 51284 by Tawa1 last updated on 25/Dec/18

If x is real, show that (2 + j) e^{(1 + j 3) x} + (2 - j) e^{(1 - j 3) x}

is also real

Commented by maxmathsup by imad last updated on 25/Dec/18

f i r s t w h a t m e a n j a n d j 3 ?

Answered by mr W last updated on 25/Dec/18

2 + i = \sqrt{5} (\frac{2}{\sqrt{5}} + \frac{1}{\sqrt{5}} i) = \sqrt{5} (\cos α + i \sin α)

= \sqrt{5} e^{α i}

w i t h α = \tan^{- 1} \frac{1}{2}

2 - i = \sqrt{5} (\frac{2}{\sqrt{5}} - \frac{1}{\sqrt{5}} i) = \sqrt{5} [\cos (- α) + i \sin (- α)]

= \sqrt{5} e^{- α i}

S = (2 + i) e^{(1 + 3 i) x} + (2 - i) e^{(1 - 3 i) x}

= \sqrt{5} e^{α i} e^{(1 + 3 i) x} + \sqrt{5} e^{- α i} e^{(1 - 3 i) x}

= \sqrt{5} e^{x} [e^{(3 x + α) i} + e^{- (3 x + α) i}]

= \sqrt{5} e^{x} [\cos (3 x + α) + i \sin (3 x + α) + \cos {- (3 x + α)} + i \sin {- (3 x + α)}]

= \sqrt{5} e^{x} [\cos (3 x + α) + i \sin (3 x + α) + \cos (3 x + α) - i \sin (3 x + α)]

= \sqrt{5} e^{x} [2 \cos (3 x + α)]

= 2 \sqrt{5} e^{x} \cos (3 x + α)

= 2 \sqrt{5} e^{x} (\cos α \cos 3 x - \sin α \sin 3 x)

= 2 \sqrt{5} e^{x} (\frac{2}{\sqrt{5}} \cos 3 x - \frac{1}{\sqrt{5}} \sin 3 x)

= 2 e^{x} (2 \cos 3 x - \sin 3 x)

= r e a l

Commented by Tawa1 last updated on 25/Dec/18

God bless you sir

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