Question

On what interval is the identity `sin^-1(sin(x))=x` valid?

Answer 1

See explanation.

Explanation:

In view of the principal value convention, x is confined to be in

`[-pi/2, pi/2]`.

This convention has to be relaxed, if inversion is governed by the

rules that, if locally bijective

`y = f(x)`,

`x= f^(-1)(y), f^(-1)(f(x))=x and f (f^(-1)(y))=y`

If this is done, the interval is `(-oo, oo)`.

This means that `sin^(-1)sin(100pi)=100pi`,

For problems in applications tn which x = a function of time, the

principal-value-convention has to be relaxed.

Having noted that there were 2.8 K viewers, I add more, to

introduce my piecewise-wholesome inverse operators for future

computers, for giving the answer as x for any `x in ( -oo, oo )`.

As of now,

calculators give answer for `sin^(-1)(sin (3600^o))` as 0. Please

wait for more details. As I have to add six illustrative graphs, I

would continue this, in my 2nd answer.

Answer 2

Continuation of my 1st answer.

Explanation:

On par with the inverse operator `f^(-1)`, I define

( trigonometric function operator sin/cos/tan/csc/sec/cot`)^(-1)`,

with `k = 0, +- 1, +- 2, +- 3, ...`.

( i ) `y = ( sin )^(-1) x = k pi + ( -1 )^k sin^(-1) x`,

`y in [ k pi + pi/2, k pi - pi/2 ]`.

Also, see
https://socratic.org/questions/defining-the-wholesome-inverse-operator-sin-1-by-y-sin-1-x-k-pi-1-k-sin-1-x-y-in#639196

( ii ) `y = ( cos )^(-1) x = 2 k pi +- cos^(-1) x`,

`y in [ 2 k pi, ( 2 k + 1 ) pi ]`.

( iii ) `y = ( tan )^(-1) x = k pi + tan^(-1) x`,

`y in ( k pi + pi/2, k pi - pi/2 )`.

( iv ) `y = ( csc )^(-1) x = k pi + ( -1 )^k csc^(-1) x`,

`y in [ k pi - pi/2, k pi ) U ( k pi, k pi + pi/2 ]`.

( v ) `y = ( sec )^(-1) x = 2 k pi +- cos^(-1) x`,

`y in [ 2 k pi, 2 k pi + pi/2 ) U (2 k pi + pi/2, ( 2 k + 1 ) pi ]`.

( vi ) `y = ( cot )^(-1) x = k pi + cot^(-1) x`,

`y in ( k pi - pi/2, k pi ] U [ k pi, k pi + pi/2 )`.

With a nuance, x = tan y is the inverse for both

`y = ( tan )^(-1) x` and `y = tan^(-1)x`.

Algebraically, the subtle difference between `( tan )^(-1)` and

`tan^(-1)` is that the second is not wholesome.

A mon avis, over centuries, the one-piece `( y in ( - pi/2, pi/2 ))`

second is preferred in Middle-School teaching, perhaps, to avoid

the complexity in discussing the piecewise-wholesome inverse. I

am sure that my operators `(tan)^(-1)` and like would be

welcomed by the relatively advanced 21st century students.

See illustrative graphs.

( i ) Graph contrasting `y = ( sin )^(-1 ) x` and `y = sin^(-1) x`:
graph{x - sin y = 0 [-20 20 -10 10]}
graph{y-arcsin x=0[-20 20 -10 10]}
( ii ) Graph contrasting `y = ( cos )^(-1 ) x` and `y = cos^(-1) x`:
graph{x - cos y = 0 [-20 20 -10 10]}
graph{y-arccos x=0[-20 20 -10 10]}
( iii ) Graph contrasting `y = ( tan )^(-1 ) x` and `y = tan^(-1) x`:
graph{x - tan y = 0 [-20 20 -10 10]}
graph{y-arctan x=0[-20 20 -10 10]}