# Thread: How do I simplify this

1. ## How do I simplify this

$\displaystyle arcsin(sin(x+\pi))$ for $\displaystyle x\in \left[\frac{3\pi}{2},\frac{5\pi}{2}\right]$

I can see that $\displaystyle arcsin(sin(x+\pi)) = x+\pi$

How does $\displaystyle x\in \left[\frac{3\pi}{2},\frac{5\pi}{2}\right]$ apply to my answer?

(I have an answer $\displaystyle x-2\pi$ and didn't understand the logic)

Thanks.

2. ## Re: How do I simplify this

Originally Posted by terrorsquid
$\displaystyle arcsin(sin(x+\pi))$ for $\displaystyle x\in \left[\frac{3\pi}{2},\frac{5\pi}{2}\right]$
I can see that $\displaystyle arcsin(sin(x+\pi)) = x+\pi$
How does $\displaystyle x\in \left[\frac{3\pi}{2},\frac{5\pi}{2}\right]$ apply to my answer?
(I have an answer $\displaystyle x-2\pi$ and didn't understand the logic
First, take note that $\displaystyle x\in \left[\frac{3\pi}{2},\frac{5\pi}{2}\right]$
is the same domain as $\displaystyle x\in \left[\frac{-\pi}{2},\frac{\pi}{2}\right]$.
That happens to be the range of $\displaystyle \arcsin(x).$
Observe that $\displaystyle \frac{3\pi}{2}-2\pi=\frac{-\pi}{2}$

3. ## Re: How do I simplify this

I can see it's domain, but I am still confused as to how I can derive an answer from that information. There is a disconnect between $\displaystyle x+\pi$ and the domain for me. How do I combine them to form an answer?

4. ## Re: How do I simplify this

Originally Posted by terrorsquid
I can see it's domain, but I am still confused as to how I can derive an answer from that information. There is a disconnect between $\displaystyle x+\pi$ and the domain for me. How do I combine them to form an answer?
To transform $\displaystyle x\in \left[\frac{3\pi}{2},\frac{5\pi}{2}\right]$ to $\displaystyle x\in \left[\frac{-\pi}{2},\frac{\pi}{2}\right]$ we use $\displaystyle x-2\pi$.

I don't know any other way to explain it.

5. ## Re: How do I simplify this

Hmm ok.What was the point of the original equation and finding $\displaystyle arcsin(sin(x+ \pi)) = x+ \pi$ then if $\displaystyle x-2\pi$ is found just by looking at the domain?

6. ## Re: How do I simplify this

Originally Posted by terrorsquid
Hmm ok.What was the point of the original equation and finding $\displaystyle arcsin(sin(x+ \pi)) = x+ \pi$ then if $\displaystyle x-2\pi$ is found just by looking at the domain?
Frankly, I do not understand the wording of the original question.
I was trying to give you a reason for the proposed answer.

I can see that $\displaystyle \arcsin(\sin(x+\pi))=\arcsin(-\sin(x))$ that is a simplification.