# opposed

1. ### When is the best time to use the spectral function as opposed to the ACF?

I know that they're equivalent ways of describing a stationary stochastic process and that they contain the same information expressed in different ways, but what kind of datasets would it be preferable to use the spectral density function? Thanks.
2. ### Circular numbers as opposed to integers and Gaussian integers?

Hi, I have almost no experience with abstract algebra (just a bit with matrices), and to start I checked out the book Numbers and Symmetry by Bernard L. Johnston. The first section introduces the Gaussian integers Z, which he contrasts with the regular integers Z. In the second chapter he...
3. ### "Fix" an element, as opposed to "consider" an element?

When doing a proof, is "considering" an arbitrary element equivalent to "fixing" an element? I don't understand the distinction, if any. For example, here's a proof from Wikipedia that uses the term "fix." So in the above proof, if I replace "fix" with "consider," is the proof logically still...
4. ### Circular functions as opposed to traditionally determined trigonometric functions?

I know they are the same thing, but my book (Schaum's Outline of Trigonometry) has a small section for "Circular Functions" where it says, among other things, that: Each arc length s determines a single ordered pair (cos s, sin s) on a unit circle. Both s and cos s are real numbers and define...
5. ### why is cos(t) -1 as opposed to 1 when P = 3pi

find the coordinates of P and the exact values of the trigonomic functions at t whenever possible. given My answers Sin(t) = 0 csc(t)= 1/0 or Undefined cos(t) = 1 sec(t) = 1/1 or 1 tan(t) = 0/1 or 0 cot(t)= 1/0 or Undefined The book Sin(t) =...