# How do you evaluate sin^-1(sin((5pi)/6))?

Aug 25, 2016

$\frac{5}{6} \pi$.

#### Explanation:

The calculator is programmed to give only the principal value for

inverse trigonometric functions.

$\frac{5}{6} \pi$ radian = ${150}^{o} \mathmr{and} \sin {150}^{o}$ is displayed as 0.5,

and inversely, $a r c \sin \left(0.5\right)$ is displayed as the principal value

${30}^{o} = \frac{\pi}{6}$ radian.

Considering sin x as a bijective function over a short interval

enclosing $x = \frac{5}{6} \pi$, the answer is $\frac{5}{6} \pi$, using that

${f}^{- 1} f \left(a\right) = a$.

If the answer is sought as the principal value in (-pi/2, pi/2), it is

$\frac{\pi}{6}$ that is yet another solution from the set of general values

$n \pi + {\left(- 1\right)}^{n} \left(\frac{5}{6} \pi\right) , n = 0 , \pm 1 , \pm 2 , \pm 3 , . .$, for $n = - 1$..