# How do you find the \arcsin(\sin(\frac{7\pi}{6})) ?

How do you find the $\mathrm{arcsin}\left(\mathrm{sin}\left(\frac{7\pi }{6}\right)\right)$ ?
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gekraamdbk
$\mathrm{arcsin}\left(\mathrm{sin}\left(7\frac{\pi }{6}\right)\right)=-\frac{\pi }{6}$
The range of a function $\mathrm{arcsin}\left(x\right)$ is, by definition.
$-\frac{\pi }{2}\le \mathrm{arcsin}\left(x\right)\le \frac{\pi }{2}$
It means that we have to find an angle $\alpha$ that lies between $-\frac{\pi }{2}$ and $\frac{\pi }{2}$ and whose $\mathrm{sin}\left(\alpha \right)$ equals to a $\mathrm{sin}\left(7\frac{\pi }{6}\right)$.
From trigonometry we know that
$\mathrm{sin}\left(\varphi +\pi \right)=-\mathrm{sin}\left(\varphi \right)$
for any angle $\varphi$.
This is easy to see if use the definition of a sine as an ordinate of the end of a radius in the unit circle that forms an angle $\varphi$ with the X-axis (counterclockwise from the X-axis to a radius).
We also know that since is an odd function, that is $\mathrm{sin}\left(-\varphi \right)=-\mathrm{sin}\left(\varphi \right)$
We will use both properties as follows:
$\mathrm{sin}\left(7\frac{\pi }{6}\right)=\mathrm{sin}\left(\frac{\pi }{6}+\pi \right)=-\mathrm{sin}\left(\frac{\pi }{6}\right)=\mathrm{sin}\left(-\frac{\pi }{6}\right)$
As we see, the angle $\alpha =-\frac{\pi }{6}$ first our conditions. It is in the range from $-\frac{\pi }{2}$ to $\frac{\pi }{2}$ and its sine equals to $\mathrm{sin}\left(7\frac{\pi }{6}\right)$. Therefore, it's a correct answer to a problem.
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enguinhispi
So:
$\mathrm{arcsin}={\mathrm{sin}}^{-1}$
${\mathrm{sin}}^{-1}\left(\mathrm{sin}\left(\frac{7\pi }{6}\right)\right)$
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.
${\mathrm{sin}}^{-1}\left(-\mathrm{sin}\left(\frac{\pi }{6}\right)\right)$
The exact value of $\mathrm{sin}\left(\frac{\pi }{6}\right)$ is $\frac{1}{2}$.
${\mathrm{sin}}^{-1}\left(-\frac{1}{2}\right)$
The exact value of ${\mathrm{sin}}^{-1}\left(-\frac{1}{2}\right)$ is $-\frac{\pi }{6}$
The result can be shown in multiple forms.
Exact Form:
$-\frac{\pi }{6}$
Decimal Form:
$-0.52359877\dots$