To begin with, I am sorry if my question is pathetic but I cannot understand why sometimes exponential involves cos and sometimes both cos and sin. I know that
e^jωt = cos(ωt) + j sin(ωt)
but further reading the book that made me ask this question, I find that
e^j(ωt + θ) = cos(ωt + θ)
Why sin is neglected in this case? Is this the same as the first?
There is probably something in the problem that leads to taking the real part. Engineering books are often sloppy on this, or you may have overlooked it. The equation as written is incorrect, but with a real part sign around the left it is correct. The use of $j$ instead of $i$ is common in engineering because they use $i$ for electrical current.