5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Students might start to feel as though trigonometric functions just go in circles. They've probably been clinging to that unit circle the same way castaways hold onto lifesavers in the middle of the Pacific Ocean. That worked in the beginning, but they're getting seasick and going nowhere. It's time to try swimming.

There might be some ups and downs, but transferring trigonometric functions onto the Cartesian plane will serve us well. To start, it's best to have students create a table of x and y values such that y = sin x. Make sure they still use radians and end up with graphs like these.

We told you. Ups and downs.

After they've been introduced to how these graphs look like, the general form y = Asin(Bx + C) + D can be introduced, where A is the amplitude, ^2π⁄_|B| is the period, C is the phase (horizontal) shift, and D is the vertical shift. This also applies to the cosine and tangent functions (except the period is ^π⁄_|B| for tangent). Students should also know that the midline is halfway between the maximum and minimum values of the graphs. The frequency is the inverse of the period.

If they feel like they're sinking, give them back the unit circle and start again, slowly working them off of it. They should feel comfortable and at least have a sense of direction when graphing these trigonometric functions. Even if your students can swim on their own, they won't get very far without a destination. (Land, ho!)

Students should be able apply the sine and cosine functions to model simple harmonic motion. Replacing physics problems about pendulums and swings with monkeys swinging from vines and bananas tied to bungee cords might help. Plus, it gets your students out of the ocean…and into the jungle, apparently.