Not really. Trigonometry in its limited, familiar form is not an area of active research. You will however find trigonometry all over various different fields of mathematical research which are active, just masked under different names and forms.
At this point it is mostly applications or generalizations of what you and I would call trigonometry. Mathematicians are now trying to explain what underlies the classes of trigonometric functions on a deeper, wider scale.
Similar to linear algebra, trigonometry is a part of mathematics which people have decided to add a full stop after.
However, just because there is a period there does not mean the chapter is over. That was just one sentence.
Mathematicians continue to examine the underpinnings of trigonometry on a deeper level, most noticeably via in real analysis and to some extent algebraic geometry. The ideas of trigonometry, on the whole, are used either as a tool, or are merely a manifestation of some deeper mathematical truth that is the object of study. Eventually those fields will also have full stops after them as mathematics gets more and more general.
Just like a good book, each sentence builds off the last one. For a current picture of mathematical research this webpage covers much of the mainstream thought.
I doubt however, that the book will ever be complete.