There are many interesting trigonometry problems in "Trigonometry for the Practical Man" by James Edgar Thompson. Here are some from Chapter 5:
From the top of a mountain three miles above sea level, the angle of depression of the
ocean horizon is found to be 2° 13' 50". Find the radius of the earth.
Answer: 3960 miles
When introducing the sum of angle equations, I practiced presenting this. Starting with introducing a right triangle inside a rectangle. (I misplaced my notes, in which I had every step clearly laid out.) In the end, it was a great proof of both equations.
I remember enjoying simplifying trig expressions.
You can find exercises all over the web. One
Here's a snippet:
Two more sources:
Since it has not been mentioned by name in the other answers, I'll say that once we model the x and y coordinates of a point on the unit circle as $\left(\cos(\theta),\sin(\theta)\right)$ (marked by a ray through the origin), I define the tangent as the slope of that ray. Since they've found the slope of a line so many times before, my students end up ...