For a complex number a find out the real and imaginary part

For a complex number a, find out the real and imaginary part of sin a. Express your result in terms of the real and imaginary part of a (i.e., a_r and a, ). Suppose we have the superposition of three sinusoidal signals as: v(t) = 3 sin (omega t + pi/6) + 4 cos (omega t + pi/4) + 2 cos (omega t - 3 pi/5) Find out the amplitude and the phase angle of v(t).

Solution

Question3 ) Ans

We have the formula sin(a+bi)=sina coshb+icosa sinhb

In our case a is complex number i.e a=a_r+ja_i

So sin(a)=sin(a_r+ja_i)=sina_r cosha_i+icosa_r sinha_i

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The above formula can be obtained using basic definition of sinx

i.e sinx=(e^ixe^ix)/2i

sin(a_r+ja_i)=(e^i(ar+jai)e^i(a_r+ja_i))/2i

if you simplify further you can get above formula

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