Halls of Ivy, I realize I said something not often said, but I intended to provoke some thought as to whether what I said is not in fact true. My point, more precisely, was that the content of so called trigonometry "courses" is so minimal or so trivial, that they do not deserve to be called courses.
As I think you yourself have pointed out elsewhere there are not really 6 essentially different trig functions, but at most one or two, or none, if you know about the exponential function. I quote you:"from the point of view of complex numbers, exponential, sine, and cosine are all the same function!"
Still perhaps I am unqualified to speak since I myself never had a trig course. We spent 6 weeks on it in high school and I stayed home those 6 weeks because school was getting so boring I needed a break.
Nonetheless, I have taught calculus for close to 40 years without much noticing the absence of that course. So perhaps I am an expert on the essential trig needed for calculus, since that is the only part I have had to learn. So I will attempt to provide a "brief trig course" here.
There are two basic trig functions called sin and cos. These are in fact inverse functions of the more natural "arclength" function. I.e. start from the point (1,0) on the unit circle and travel counterclockwise along the circle, to the point (x,y). Then for 0 <= y <= 1, the arclength traveled is a function of y, and sin is the inverse of this function. I.e. sin(t) = the y coordinate of the point reached after traveling along a circular arc of length t, in the counterclockwise direction. cos(t) is the x coordinate of the same point. These latter two functions are defined for all real t, even negative, by going on around the circle more than once, or by going clockwise for negative t.
The Pythagorean theorem implies then that for all t, cos^2(t) + sin^2(t) = 1.
This is the first big trig fact, and the most important one. Draw a picture and convince yourself of this before continuing.
The other trig functions apparently do not exist in all parts of the world, since they are just formed from dividing these, but we call sin/cos = tangent = tan, cos/sin = cotangent = cot, 1/sin = cosecant = csc, and 1/cos, = secant = sec.
Tangent is convenient because, since cos = 0 at -pi/2 and pi/2, tan goes to infinity at those points and provides a nice example of a one one correspondence between the finite interval (-pi/2, pi/2) and the whole real line.
You should now stop and graph all three functions, sin, cos, and tan and look at their graphs. You notice that they are all "periodic", i.e. repeating cyclically, in the sense that f(t+2pi) = f(t), for all t, and all trig functions f. The graphs also reveal that sin(t) = cos(t-pi/2).
A basic fact is that it is very hard to compute values of these functions. E.g. I myself do not know the value of cos(1). There are a few simple arguments whose trig values can be computed by geometry however and it helps to know these. By drawing coordinate triangles in the circle, you can see that the following versions of the trig functions are also valid:
We will say the arclength t on the unit circle cuts out the "angle t" at the origin, in terms of "radian" measure. Then if we draw a right triangle with acute angle t, we have sin(t) = opposite/hypotenuse, cos(t) = adjacent/hypotenuse, tan(t) = opposite/adjacent, etc..., where these ratios are of lengths of the correspondingly named sides of the triangle. Stop nopw and look at a picture of a unit circle to convince yourself these are the same as the original definitions of the trig functions. This approach is called "triangle trig."
From elementary triangle geometry you can deduce the values of the trig functions at pi/2. pi/3, pi/6, pi/4, and you should do so. E.g. cos(pi/2) = sqrt(2)/2 = sin(pi/2), and cos(pi/3) = 1/2, cos(pi/6) = sqrt(3)/2.
You now know about as much or more trig as the average entering calculus student, but I will continue with the "honors course" in trig. This consists of a few useful formulas which most students do not remember from their trig course.
Double angle formulas: sin(2t) = 2sin(t)cos(t). (A recent question on this site reveals that some current students of trig do not know this.)
cos(2t) = cos^2(t)-sin^2(t).
Once you know those you can bootstrap up to remembering their generalizations (addition formulas):
sin(s+t) = cos(s)sin(t) + sin(s)cos(t).
cos(s+t) = cos(s)cos(t)-sin(s)sin(t). Do you see the similarity?
These can be proved using triangle geometry, but few students know how to do this, I would guarantee. If you want a challenge, draw the picture for the trig values on the unit circle for the angle s+t, then rotate the picture down so that the arc of length s extends to one side of the point (1,0) and that of length t extends to the other. Then use pythagoras I guess, as I have not done it lately. Note these formulas imply the fact sin(t) = cos(t-pi/2), e.g.
This is plenty of trig for the first two courses in calculus. The only other formula I think you might need later, in vector situations, is the generalized Pythagorean theorem, called the "law of cosines". In a triangle with sides R,S,T and opposite angles r,s,t, this says that
R^2 + S^2 - -2RScos(t)= T^2. Notice if T = pi/2, this is Pythagoras. (I had to look this up myself.)
In Edwards and Penney, Calculus, there is a 6 page review of trig in the appendix, including a page of exercises, and then less than 3, pages of review of formulas from geometry and trig. That is more than you need for calculus.
Good luck.
) and was actually surprised how SIMPLE math functions were. I was used to having a function... DO stuff... and not just return a value. So it was really like a step down.
