# My calculator displays sin^(-1)sin 45^o=45^o, sin^(-1)sin 135^o=45^o and sin^(-1)sin 405^o=45^o.How do you convert this answer from the calculator to the specific starter value? for similar other trigonometric forms?

Oct 15, 2016

The algorithm, of the program that is executed, chooses the

principal value

${x}_{P} \in \left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$ against the input value x for ${\sin}^{- 1} \sin x$

Obviously, if you see x as the answer, using f(-1)f(x)=x. do not use the

device. Such an answer is needed, when x radian = x(time), as in the

wave equation, y = sin x, where x = ct.

The general value of x is

$180 k + {\left(- 1\right)}^{k} {x}_{P} , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

I have used here $\pi$ radians = ${180}^{o}$

For example, the input $x = {405}^{o}$, the answer displayed is

${x}_{P} = {45}^{o}$.

$405 = 180 k + {\left(- 1\right)}^{k} 45$, for k =2..

For ${\cos}^{- 1} \cos x$,

$x = 360 k \pm {x}_{P}$, for suitable k.

For example,for the input $x = {225}^{o}$, the answer displayed is

${x}_{P} = {135}^{o}$.

$225 = 360 k \pm 135$, for k = 1.

for ${\tan}^{- 1} \tan x$,

it is same as for ${\sin}^{- 1} \sin x$.