Many great statements about why study mathematics in general have come out of this discussion. But what about trig in particular? If I were to teach trig, I imagine that the first day I would start with a graph of something like day of year versus number of hours of daylight. Then ask how is this different from graphs that you saw in algebra. It is pretty obvious that this is a fundamentally different creature from a polynomial function. Then segue into the idea of families of functions. Like there are plants and animals, vertebrates and invertebrates, in mathematics there are different kinds of functions. The purpose of trig is to study one of these families. What questions do you want to know about any function? Discuss what sorts of issues came up with the algebraic functions. Remember the square root? It lead to irrational numbers. So do trig functions. They are part of a large family called transcendental functions.
Throw in a little history. Mention some (diverse) applications. Tie it in to what they already know. Give an idea about where this course leads to. That to me is "motivation."
Then, roll up your sleeves and teach them some trig.