Some insights
Some Trigonometry
You know this already, right?  You've seen the trig functions defined as power series, and you know the "standard" definitions of cos as adjacent over hypotenuse and so forth.  But do you know what co means in cosine?  And do you know what arc means in arcsine?  And do you know what to do if you unexpectedly need a trig identity while you're at the beach, far from your CRC handbook, and you can't remember it?

Basic Definitions

Let's take a quick look at the unit circle, and see what the basic trig functions are.  They're all just lengths, of course, when the radius of the circle is one; in that case there are no ratios to worry about.

There are two angles marked in the diagram,
theta and phiphi is the complementary angle to theta, and the sine, cosine, and tangent of phi are the complementary functions for theta:

   sine (phi) = sine of the complement of theta = cosine (theta)
   tangent (phi) = cotangent (theta)
   secant (phi) = cosecant (theta)

What about the inverse functions?  The arclength of the angle theta is equal to theta.  So, the arclength associated with the length of the sine line, for example, is the same as the angle associated with it.  Hence,

   arclength associated with sine value of x = arc of the sine =  arcsine (x) = theta = sine-1 (x)

which is why the inverse trig functions are all called "arc...", unlike every other inverse function in the universe.

Double-Angle, Angle-Sum, and Half-Angle Formulas

You learned them.  I learned them, too.  If you're like me you can't remember them.  But, you don't need to, because they're all just compositions of rotations, and you can derive whatever you need from the rotation matrix with very little effort.  And so, all you really need to remember is the form of a circular rotation.

A rotation of the X-Y plane by θ is given by the matrix:

If we rotate by θ and then by φ the total rotation is by θ+φ, and it's given by the matrix

But it's also just the composition of the rotation matrix for θ with the rotation matrix for φ:

So, to find any of the terms in the combined rotation matrix, we just need to carry out the matrix multiplication.  We can read off directly:

That was a one step derivation.

The double-angle formulas are just special cases of these, so we can also see immediately that

If you need the half-angle formulas, you can just substitute φ/2 for θ in the double-angle formulas.  The half-angle cos formula is immediate:

The sine formula takes some algebra, but is easily derived from the cos formula:

Again, my point is not that you want to go through this exercise whenever you do any algebra.  Rather, my point is that if you're stuck for an identity, don't give up -- it's often not that hard to figure them out from scratch.  In fact, there's less to remembering the derivation than there is to remembering the finished formulas.

Other Trigonometric Identities

There are lots of trig identities besides the angle-sum formulas, of course, and they're almost all hard to remember unless you use them every day.  Obviously, if you're near a reference book or a computer, if you need one you just look it up.  But if you're stuck,, you can usually pull the simple ones off a picture using Pythagoras's theorem, and a number of the others are very easy to derive from the basic ones.  In fact, I, at least, find it a lot easier to recall the general approach to the derivations than I find it is to recall the exact formulas.  Here are two examples, one trivial, one a little more complicated.

Suppose you have the sine, and you need the tangent.  Just draw a picture, label the sides so that one of them has the length you know (the sine, in this case), and you can read off the other functions directly:

tangent from sine

But that's pretty trivial, and you could do it in your head.  Here's a harder one.

Suppose you need the derivative of the arctangent, and you just can't recall what it is.  What to do?

Well, you surely recall the derivatives of sine and cosine, even if you can't recall any others.  So rewrite the problem in those terms:

and now just push a "d" operator through it, and use the chain rule and quotient rule to work it out:


Once the "d"s have gone all the way through, you just need to get the right one on the bottom.  Multiply through by and you're done:


Oh -- but here we've got a cos(theta) in the result, and all we know is that x = tan(theta).  Just draw a triangle, read off the conversion, and plug it in:

cosine from tangent
And we get


Isn't that faster than driving all the way home for a CRC handbook?

Page created in 2004.  Minor changes to diagrams, 11/14/06