Prove Identity: Alternatives to Derivation

[icystrike]

Homework Statement

[PLAIN]http://img812.imageshack.us/img812/4068/deriveidentitybydesmond.jpg

In fact i came up with this identity just wondering if there is alternative way to prove it.

Homework Equations

The Attempt at a Solution

 

[Mentallic]

You made a slight typo, it should be $$\sin^{-1}\left(\frac{a}{r}\right)$$ at the end.

To make things clearer, you should let a/r=x and state the restriction on x for computable real values.

Ok anyway to the point... You're looking for an alternate way to proving this identity? What is the first way you proved it?

 

[Tedjn]

Without investigating further, do you require r = ±a? Because the domain of inverse sine is between ±1, but you have both a/r and r/a as arguments.

 

[Mentallic]

Without investigating further, do you require r = ±a? Because the domain of inverse sine is between ±1, but you have both a/r and r/a as arguments.

Well that's why as I said, it must've been a typo such that the inverse sine on the RHS is a/r.

 

[icystrike]

Well that's why as I said, it must've been a typo such that the inverse sine on the RHS is a/r.

Yes sorry for my typo...
then i guess -1<=x<=1

I derived this identity when someone told me to integrate a small area bounded by the upper quadrant of circle and x=a and y-axis ... then i derive two method ... one is by substitution and integrate with repect to theta and next is by geometric means (area of part of circle minus a small triangle) Ignore my proof.. I more interested in a more generalized identity or theorem that can explain this

 

[Mentallic]

That's a little complicated, but fair enough

Let $$a/r=x$$ and $$\sin^{-1}x=A$$, now look at the LHS - you should obviously know your double angle formulae, mainly $$\sin(2A)=2 \sin A \cos A$$

Do you know how to find $$\cos\left(\sin^{-1}x\right)$$?

After doing this you'll notice that your identity can be simplified greatly.

 

[icystrike]

Ok! I've proved... i wasnt quite aware of the existence of the list of identities for arc function ... ( I am wondering at which level you will be more exposed to the identities of the inverse function?)

 

[Mentallic]

Identities for arc functions? If you mean $$\cos(\sin^{-1}x)=\sqrt{1-x^2}$$ and such as being identities, well, frankly I'm quite surprised you haven't learned them already.