# Day 64: Composite Trig Functions

I asked my precalculus students this, “what do you think happens when you add, multiple or compose trigonometric functions with functions from other families? Will the combined functions be periodic?  What type of shape might you expect?”  I had four explorations ready for them to test out conjectures about what happens.  I prefaced the experience by encouraging them to ask “What if? “How?” “Why?” questions as they worked through the explorations…and some actually did.

Initial student comments around the beginnings of the explorations: “It’s a sine function on a slant!” “How are we ever going to find the equation?” “Oh slam…you’re right!”  It was so engaging for me as the teacher listening to the conversations, the conjecturing, and the testing/revising until my kiddos found an equation that actually did what they wanted.  They also looked at predicting what an equation would look like graphed before they graphed it.  Lots of dendrites growing today!

Here is one that combined sine, cosine and tangent.  When asked how each showed up in the final graph, my students were able to point out exactly what they saw and why.

Finally, I had one student on his computer using Desmos with the exploration.  And its available on the iPads for free.  Although I am a big fan of the TI Nspires, I do wonder if Desmos might be a good alternative for some settings.  Have to ruminate on that!