Trigonometry - Graph of Tan lesson

Hi, I've really enjoyed using Khan Academy and I have learned a ton of trig very rapidly (for me). I've found each lesson easy to follow and clear, and I also like that I can pause and guess where things are going, or watch each concept get developed step-by-step if I can't predict.

The exception is the lesson where the tangent of theta is graphed. Sine and cosine and their intersection points were all great, but this video jumps right to the fact that this is a periodic function reaching infinity every pi radians, all at once, without letting us see that concept develop. First pi/4's tangent is graphed, then negative pi/4, and it looks like a straight line is forming, but we don't test further to see for ourselves that's not what's happening--we are simply told, and the graph is drawn as a curve without our understanding why. Please fill in the gaps hear and lead us to the solution, like you usually do. Thank you!