> Discuss what sorts of issues came up with the algebraic functions. Remember > the square root? It lead to irrational numbers. So do trig functions.
So do linear functions, you know. I guess you mean that the square root of an *integer* can be irrational. On the other hand, the cosine of (2*pi) is 1, very rational. I think you will have to phrase your discussion with care.
> They > are part of a large family called transcendental functions.
True, but to prove that "sine" is not an algebraic function, for example, is not for a high school course. On the other hand, any student of trigonometry ought to be able to understand that cos(arctan x) = 1/(sqrt(1+x^2)), an algebraic function defined for all real x. Most calculus students have trouble with this one, because it turns up too suddenly, in connection with derivatives.
> Throw in a little history.
Now you're talking! To carry back the formula for cos(A+B) to ancient times is, however, not easy. There is a proof of the equivalent attributed to Ptolemy, but it requires quite a bit of knowledge of Euclid to follow it. But even if a trig class doesn't get the full story, it is very good to tell as much of the chronological story as possible. On the other hand, historical trigonometry is not about periodic functions, but is confined to angles inscribed in circles somewhere, i.e. angles that can occur in triangles; and the extension of the trig functions to larger domains is more recent. Here, history can be mischievous if one is not careful. Old fashioned trig books would give a definition of sine as a ratio in a right triangle, hence for acute angles only, and then extend the definition to the other quadrants by a woefully long list of adjustments. From a strictly historical point of view that may be more or less the way it happened, but it is better to give the y/r definition as soon as possible, and *use* it, either in talking about complex numbers or about polar coordinates in general. And then *use* complex numbers for something as soon as possible, e.g. De Moivre's theorem for roots of unity, or any other complex number, according to how much time your curriculum gives you for such impractical things.
Mention some (diverse) applications. Tie it in to > what they already know. Give an idea about where this course leads to. That > to me is "motivation." > Here is some motivation for trig functions, apart from solving triangles, and apart from more sophisticated applications such as alternating current and Fourier Series descriptions of musical tones:
Roots: How come there are two square roots of a positive number but only one cube root? How come there are *no* square roots of a negative number, but one cube root? What about fourth roots? nth roots?
Directions: In Manhattan one tells a stranger how to get somewhere by saying it is five blocks north and two blocks west from where you are. In an Iowa cornfield, what do you tell him? And then, when the building developer spoils the landscape with housing or office buildings, how do you change the Iowa description back to the New York style?
Graphs: Explain the cardioid microphone, and why it is that people can see so much of the TV screen from even fairly extreme positions to the right and left of the selfish people who had turned the set on for themselves only.
> Then, roll up your sleeves and teach them some trig. >