Unfortunately I don't have any way to draw pictures on this computer, otherwise I could give a more helpful answer, but I have thought about this and I would start with a diagram a little like the following from wikipedia - but initially only with cos, sin and tan (and maybe cotangent):
I would start by motivating sine. Depending on how concrete the students need things to be (and some need more than others) I would imagine the unit radius line as (say) a crane or robot arm. The sine of the angle is then the height of the arm above the horizontal. This is true for all angles even ones outside the 0 - 360 degree range.
When the arm goes below the horizontal, the sine is (obviously) negative because you are going below the line. It is easy to see that sine is positive from 0 to 180 and then negative from 180 to 360.
Cosine is then the distance projected onto the horizontal by the line (eg the length of shadow cast by the crane or how far the robot arm stretches at that angle). Again this is true for all angles. You can see it is periodic and will become negative as the angle passes 90.
Tangent is less obvious but it should be clear that it grows without bound as the angle approaches 90 (and you can draw some pretty long tangents to show this). The range -90 to 90 is fairly clear. Obviously you can extend the definition, but the intuition requires a little more work.
Sec and cosec are likely to be for more advanced students but you can see how they have their own definition as well as being multiplicative inverses of cosine and sine. Sin^2 + cos^2 = 1 is hopefully obvious if you know your pythagoras (and this is a way of re-using that formula and reinforcing it), other identities can be drawn out geometrically.
Now, I suspect that this works much better for some students than others. Some people really don't have or like to use a pictorial way of thinking about something, but I think this is useful.