NavList:

You wondered what the "caret" stands for in something like x^2. It's an early online way of expressing exponentiation. No doubt, you've seen it before, but there are other NavList readers who certainly do not know what it means. It's exponentiation, so x^2 is identical to "x squared" which can also be written
   x²
when using a Unicode (UTF-8) character --this is where the superscript "2" is a separate character all by itself, or, also,
   x²
when the normal character "2" is shifted up and decreased in size by markup language (the "ML" in HTML). The "caret" dates from the era before either of these more "modern" notations existed. Another old way of expressing exponentiation was a double asterisk **, as in x**2. Although the modern Unicode and ML versions look very nice, the caret notation remains valid and critical because it is recognized semantically in computer code and especially spreadsheet formulas (in a spreadsheet, if you put "2" in cell A1 and "3" in cell A2, and then the formula "=A1^A2" in cell A3, then A3 will display the value two cubed... 8).

Back to Bill's explanation, starting with the usual fundamental formula or law of cosines, we can immediately decide it's a right triangle which then implies:
  cos c = cos a · cos b,
but we know that for small angles, when the angles are expressed as pure numbers (also known as radians), any cosine is given approximately by
  cos x = 1 - x²/2,
and then following a little algebra, we find that c² = a² + b². The part that's unexpected is that this works for angles of up to several degrees. That was the 3:4:5 issue I brought up previously.

Frank Reed