# Thread: Exact value of arcsin(2/3)-arcsin(-2/3)

1. ## Exact value of arcsin(2/3)-arcsin(-2/3)

What is the Exact value of arcsin(2/3)-arcsin(-2/3)?

2. ## Re: Exact value of arcsin(2/3)-arcsin(-2/3)

Use WolframAlpha!

It says $2arcsin \frac {2}{3}$ There is no convenient form of the exact value of the question. Are you looking for ten significant figures? Nine? Eight? Twenty? Use WolframAlpha! :P

3. ## Re: Exact value of arcsin(2/3)-arcsin(-2/3)

I guess it should be in terms of pi or square root.

4. ## Re: Exact value of arcsin(2/3)-arcsin(-2/3)

Originally Posted by yeoky
I guess it should be in terms of pi or square root.
I'm not seeing any wizardry that would allow this to be evaluated exactly. Are you sure you have the problem correct?

5. ## Re: Exact value of arcsin(2/3)-arcsin(-2/3)

$\displaystyle \sin^{-1}(x) = \sum_{k\ge 0} \dfrac{x^{1+2 k} \left(\tfrac{1}{2}\right)_k}{k!+2 k\cdot k!}$

Plug in $x = \dfrac{2}{3}$

The $\left(\tfrac{1}{2}\right)_k$ is the Pochammer symbol, meaning falling factorial.

6. ## Re: Exact value of arcsin(2/3)-arcsin(-2/3)

Originally Posted by SlipEternal
$\displaystyle \sin^{-1}(x) = \sum_{k\ge 0} \dfrac{x^{1+2 k} \left(\tfrac{1}{2}\right)_k}{k!+2 k\cdot k!}$

Plug in $x = \dfrac{2}{3}$

The $\left(\tfrac{1}{2}\right)_k$ is the Pochammer symbol, meaning falling factorial.
that's some wizardry all right!

7. ## Re: Exact value of arcsin(2/3)-arcsin(-2/3)

Originally Posted by LimpSpider
Use WolframAlpha!

8. ## Re: Exact value of arcsin(2/3)-arcsin(-2/3)

I wonder if this is what they are after.

$\arcsin(\frac 2 3) - \arcsin(- \frac 2 3)=2 \arcsin(\frac 2 3)$

$y = 2 \arcsin(\frac 2 3)$

$\sin(y) = \sin\left( 2 \arcsin(\frac 2 3)\right)=$

$2 \sin\left(\arcsin(\frac 2 3)\right)\cos\left(\arcsin(\frac 2 3)\right)=$

$2(\frac 2 3)(\frac {\sqrt{5}} 3)=\frac {4 \sqrt{5}} 9$

$\sin(y)=\dfrac{4 \sqrt 5} 9$

$y = \arcsin\left(\dfrac{4 \sqrt 5} 9\right)$

It is the correct value.

9. ## Re: Exact value of arcsin(2/3)-arcsin(-2/3)

Complex Solution