| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
A rational function is a quotient of polynomials. tan(x/2) can be written as a rational function of sin x and cos x, namely: sin x tan x/2 = --------- . 1 + cos x Question: Can sin(x/2) be written as a rational function of sin x and cos x? Same question for cos(x/2).
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 399.1 | METOO::YARBROUGH | Tue Dec 10 1985 09:18 | 4 | ||
No, because the set of zeroes of sin(x/2) is 2n*pi, while any polynomial in sinx and cosx has zeroes at n*pi intervals, thus any rational polynomial in sinx and cosx has zeroes at the same intervals. The polynomial in the denominator can introduce poles, but not zeroes, to the function. | |||||
| 399.2 | TOOLS::STAN | Tue Dec 10 1985 14:06 | 6 | ||
Sounded great :-) until I thought about it :-( for a while. "any polynomial in sinx and cosx has zeroes at n*pi intervals" Wrong. The polynomial sin x + cos x + 17 does not have zeroes at n*pi intervals. | |||||
| 399.3 | ADVAX::J_ROTH | Tue Dec 10 1985 15:55 | 5 | ||
I think response .1 still stands. sin(x)+cos(x)+17 does have periodic zeroes in the complex plane... - Jim | |||||
| 399.4 | HARE::STAN | Sat Jan 18 1986 20:34 | 5 | ||
The answer is no. Consider the smallest angle in a 3-4-5 right triangle. Call this angle x. A simple calculation shows that both sin(x/2) and cos(x/2) are irrational. However, sin x and cos x are rational. If sin(x/2) were a rational function of sin x and cos x, then it would be rational too. A contradiction. | |||||