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

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Answer 1 · 2016-10-15T04:57:19

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

principal value

$x_{P} \in [- \frac{π}{2}, \frac{π}{2}]$ $x_P in [-pi/2, pi/2]$ against the input value x for ${sin}^{- 1} sin x$ $sin^(-1)sin x$

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

$180k+(-1)^k x_P, k =0, +-1, +-2, +-3, ...$

I have used here $pi$ radians = $180^o$

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

$x_P=45^o$ .

$405=180k+(-1)^k 45$ , for k =2..

For $cos^(-1)cos x$ ,

$x= 360k+-x_P$ , for suitable k.

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

$x_P=135^o$ .

$225=360k+-135$ , for k = 1.

for $tan^(-1)tan x$ ,

it is same as for $sin^(-1)sin x$ .