arcsin(sin(x)) ?

**circuscircus** · September 18th 2007, 08:56 AM

How do I know when $arcsin(sin(x))$ is

$x$
$\pi - x$ or
$2\pi - x$ ?

**Krizalid** · September 18th 2007, 10:10 AM

Originally Posted by circuscircus

How do I know when $arcsin(sin(x))$ is

$x$
$\pi - x$ or
$2\pi - x$ ?

Set $\alpha=\arcsin(\sin x)\implies\sin\alpha=\sin x$

**ThePerfectHacker** · September 18th 2007, 01:13 PM

The function $\sin^{-1} (\sin x)$ is very interesting. But it gives back $x$ when $|x| \leq \frac{\pi}{2}$ .

**circuscircus** · September 18th 2007, 07:37 PM

Yea but when it's like

$arcsin(sin(3))$

$arcsin(sin(4))$

$arcsin(sin(5))$

I think the first one is $\pi-3$ , second one is $\pi-4$ but last one is $\2pi-5$ (I'm going by memory from the lecture but you get what I'm saying like when it is outside that |pi/2| bound, it gets weird but I'm going to get tested on that

I don't understand why it is those values...

**ThePerfectHacker** · September 18th 2007, 07:45 PM

Have a look at this picture.
(I might find its Fourier series if I am bored. It looks nice).

**circuscircus** · September 18th 2007, 07:50 PM

I'm sorry...I don't understand graphs too well...could you explain what it means? I think I understand like have the brief idea but I can't translate the stuff I see on the chart to like pi-3, 2pi-5, etc...

**ThePerfectHacker** · September 18th 2007, 07:56 PM

Originally Posted by circuscircus

I'm sorry...I don't understand graphs too well...could you explain what it means? I think I understand like have the brief idea but I can't translate the stuff I see on the chart to like pi-3, 2pi-5, etc...

Let $y=\sin^{-1} (\sin x)$ . It means if you got at a certain point on the x-axis say $x=\frac{\pi}{2}$ then the y-value is the output you get if you subsitute $x$ for $\frac{\pi}{2}$ in $\sin^{-1}(\sin \frac{\pi}{2} )$ . The reason why I graphed it is so that you can just find all the values by just looking without calculating anything.

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