Proving sin^0 (x): Is it 0, x, or Something Else?

[PhysForumID]

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

 

[Number Nine]

$$sin^{0}(x) = (sin0)^{0} = 1$$
Is n a natural number? Then start with n = 1.

 

[PhysForumID]

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

 

[Number Nine]

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

$sin^{2}(x)$ is shorthand for $sin(x)sin(x)$, and so on for arbitrary n. A value raised to the power of 0 equals 1 due to the fact that $x^{n} = x*x^{n-1}$, so...
$$x^{0} = x*x^{-1} = x*\frac{1}{x} = 1$$

and yes n is a natural number but starting from 0

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?

 

[PhysForumID]

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

 

[spamiam]

Can you show that $\sin^{n}(x) > \sin^{n+1}(x)$ 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 $\sin^2(x)$ to mean $(\sin(x))^2$ is very unfortunate. It is often the case that $f^2(x)$ is taken to mean $f(f(x))$ as you had thought, PhysForumID. This is almost always the case with the exponent -1, since $f^{-1}$ usually denotes the inverse of f with respect to functional composition, not multiplication. One great confusion people often have while learning trigonometry is that $\sin^2(x) = (\sin(x))^2$, but $\sin^{-1}(x) \neq (\sin(x))^{-1}$. Rather $\sin(\sin^{-1}(x))=x$, since here the exponent refers to functional composition and not multiplication.

Zero isn't a natural number. This is me being pedantic, of course, and you can still begin with n=0 if you like.

There is no consensus on whether or not 0 is a natural number. From http://en.wikipedia.org/wiki/Natural_number" :

Including 0 is now the common convention among set theorists, logicians, and computer scientists. Many other mathematicians also include 0, although some have kept the older tradition and take 1 to be the first natural number.

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."

 

[Redbelly98]

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