A natural way to talk about trig functions is to model using a wheel...in this case a ferris wheel. Here is a problem my students have worked on recently.
Here is an animated GIF of a classical optimization problem: Regiomontanus' angle maximization problem. A similar problem appeared on our final exam yesterday. My students commented that for a variety of reasons the "optimum" solution isn't so optimum, especially if you are watching TV.
I spent this year in an effort to transition my students from a the tradition lecture, example, home work model they were comfortable with to problem based, discussion model. The transition also involved moving from away from a traditional text book for students to follow and to problem sets designed to invite students to develop their own mini text books for reference. Looking back at their work from the fall, its great to see how far they have come.
"Your only as good as your record collection." -DJ Spooky