This is a continuing series of posts about how I approach topics in Pre-Cal They represent a sample of what I get the most questions about which always seem to start "how do you..."
Students understand the unusual nature of a trigonometric equation. Students understand there is a base set of solutions on the domain 0 to 2pi, but an extension of that domain extends the solution set. Students understand that it is possible to make a statement that summarizes all the terms in the infinite series of solutions. Students understand that the frequency of the trig functions relates to the way solutions repeat along the domain.
We don't start with algebra. We start with the concept of an inverse trig function. For convenience, I stick with angles on the unit circle. My first request is that given a particular x or y coordinate from the unit circle, can you find all the angles associated with that coordinate? We condense that statement into math notation, sin^-1 x or arcsin x, for example. The idea that multiple answers are possible is interesting, and lays the foundation for the algebra later.
Next, we look at graphs. I hand out a set of trig equations.
To keep things simple I have them graph the even set or odd set on a restricted domain of 0 to 4.5pi, recording every intersection they notice with a sketch of what they saw. This continues a big algebra idea, every equation has a graph that validates the algebra. For more complex algebra, I always like to start with the graph.
For some of these equations there are many intersections, for others there is but one. Some comments on how weird tangent is, and questions on how to type sin^2 x.
Now the interesting question. Why so many answers? What happens if we widen the domain restriction? What if we removed it entirely? Many correctly guess that there are far more solutions than we could ever write down.
But is there something to this? Is there a way to condense a small infinity like this into something a little more manageable? We take a closer look at just what numbers appear as solutions. Are they random or regular? How might we predict the next number?
After the graphing exercise I graph another random trig equation. We focus on the first two positive solutions. I name them Solution Zero. We discuss the frequency of the function in question. I take our Solution Zero and add one repetition of the frequency to those numbers. Amazingly, the next number in the solution sequence comes out. Minds blown.
Eventually we discuss that an infinite set of solutions condenses to Solution Zero + n[frequency] both in radians and degrees. Finding Solution 1000 is an easy task now. It's fun to ask about how we would determine solutions to the left of zero, many determining that n = -1 would do the trick.
Lastly I demonstrate the algebra necessary to find these solutions without a graph, validating the need for an inverse trig function.
We build from there and talk about squared functions and how the scope of Solution Zero is much larger (base 4 instead of 2). All of it a lovely dance back and forth between algebra and graphs, avoiding impractical (and graphically dubious) equations used by textbooks to convince somebody (ANYBODY) that kooky trig identities are a thing real people use.
We discuss Trig Equations in the early parts of the second semester. I look for student understanding of both the graphical and algebra aspects of the solution sets. Students validate whether a graph accurately represents a trig equation and determine solutions beyond Solution Zero.