Trigonometry
posted by steve .
complex numbers

They must be imaginary, I don't see them

just as real numbers can be located on the number line, complex numbers can be located in the plane, because they have a real part and an imaginary part. So,
a+bi
is located in the plane by a pair of coordinates: a units along the horizontal axis, and b units vertically from there.
Not sure how a long discourse here will be any more useful than the discussion in your text. Or, do a web search, and you will find many articles explaining them. 
The reason they come up in trig is that any point in the plane can be located at some distance from the origin, and in some direction, given by an angle θ. All your normal trig functions can be applied to θ.
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