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?

There are two angles marked in the diagram, and . is the

sine () = sine of the complement of = cosine ()

tangent () = cotangent ()

secant () = cosecant ()

What about the inverse functions? The

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

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

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.

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:

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() in the result, and all we know is that

And we get

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