Question

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?

Answer 1

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

principal value

`x_P in [-pi/2, pi/2]` 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

`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`.