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Sin0 (x)

  1. Oct 15, 2011 #1
    What would sin^0 (x) mean? sin^n (x) means using the sine function 'n' times on x, so what does it mean to use it zero times? does sin^0 (x) then equal 'x' or '0' or... ?

    The context of this question is that I have to prove that:
    I_n = integral from zero to pi/2 of sin^n (x) with respect to 'x'

    I am proving this by induction starting with n=0, assuming true for n=n and showing it is true for n=n+1
     
  2. jcsd
  3. Oct 15, 2011 #2
    [tex]sin^{0}(x) = (sin0)^{0} = 1[/tex]
    Is n a natural number? Then start with n = 1.
     
  4. Oct 15, 2011 #3
    sorry maybe I've got more than that confused in my head... I always understand sinn (x) to mean you use the function 'sine' 'n' times on x, rather than take sin(x) and multiply it by 'n'... am I wrong there? Surely not because sin(sin(pi/2)) >< {sin(pi/2)}^2

    and yes n is a natural number but starting from 0
     
  5. Oct 15, 2011 #4
    [itex]sin^{2}(x)[/itex] is shorthand for [itex]sin(x)sin(x)[/itex], and so on for arbitrary n. A value raised to the power of 0 equals 1 due to the fact that [itex]x^{n} = x*x^{n-1}[/itex], so...
    [tex]x^{0} = x*x^{-1} = x*\frac{1}{x} = 1[/tex]

    Zero isn't a natural number. This is me being pedantic, of course, and you can still begin with n=0 if you like. Can you elaborate on what you're trying to prove? What in the integral supposed to equal?
     
    Last edited: Oct 15, 2011
  6. Oct 16, 2011 #5
    wow I have no idea how I got this far in uni making that mistake about what sin^2(x) was... thanks number nine :)

    and sorry, that was my mistake for saying it was a natural number. n = {0,1,2...}

    the integral is given and we have to show that I_0 > I_1 > I_2 > .... etc
     
  7. Oct 16, 2011 #6
    Can you show that [itex]\sin^{n}(x) > \sin^{n+1}(x)[/itex] for all n? Once you do, can you see how to use this to solve the problem?

    Also, I'd like to say that I think the notation [itex]\sin^2(x)[/itex] to mean [itex](\sin(x))^2[/itex] is very unfortunate. It is often the case that [itex] f^2(x)[/itex] is taken to mean [itex] f(f(x))[/itex] as you had thought, PhysForumID. This is almost always the case with the exponent -1, since [itex]f^{-1}[/itex] usually denotes the inverse of f with respect to functional composition, not multiplication. One great confusion people often have while learning trigonometry is that [itex] \sin^2(x) = (\sin(x))^2[/itex], but [itex] \sin^{-1}(x) \neq (\sin(x))^{-1}[/itex]. Rather [itex] \sin(\sin^{-1}(x))=x[/itex], since here the exponent refers to functional composition and not multiplication.

    There is no consensus on whether or not 0 is a natural number. From http://en.wikipedia.org/wiki/Natural_number" [Broken]:
    You can use either convention as long as you're consistent. If you really want to be unambiguous, you can say "non-negative integers" and "positive integers."
     
    Last edited by a moderator: May 5, 2017
  8. Oct 17, 2011 #7

    Redbelly98

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    Moderator's note: thread moved from "General Math" to "Homework & Coursework Questions". Rules for homework help are in effect.
     
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