Describe what the Eulers equation ej omega t cos omega t j

Describe what the Euler\'s equation e^j omega t = cos (omega t) + j sin (omega t) communicates?

Solution

(e^{j\\theta}=cos (\\theta) + j sin (\\theta) \\) is very useful not only in obtaining the rectangular and polar forms of complex numbers, but in many other respects as we will explore in this problem.

(a) Carefully plot \\(x[n]=e^{j\\pi n} for -\\infty < n < \\infty\\) . Is this real or a complex signal?

(b) Suppose you want to find the trigonometric identity corresponding to \\(sin(\\alpha) sin(\\beta)\\)

a)

e^jn=cos(n)+jsin(n)

sin(n)=0 for every -<n<

hence

e^jn=cos(n) is always real since there is no imaginary part isin(n)

b)

e^j=cos()+jsin() -----> 1

e^-j=cos()-jsin() -----> 2

from above

subtracting 2 from 1

e^j-e^-j = 2jsin()

sin()=(e^j-e^-j)/2j

hence

sin()sin()= [(e^j-e^-j)/2j][(e^j-e^-j)/2j]

= [e^2j+e^-2j-(e^j-j+e^-j+j)]/(-4)