I would like to show that sin − 1 is an increasing function. In other...

Ciara Mcdaniel

Ciara Mcdaniel

Answered

2022-07-10

I would like to show that sin 1 is an increasing function. In other words, I want to show that
x 1 x 2 sin 1 ( x 1 ) sin 1 ( x 2 ) .
I fail to see how to start this. Also, if possible, I would like to show this without using other functions (e.g. sin) known to be increasing/decreasing.

Answer & Explanation

Elias Flores

Elias Flores

Expert

2022-07-11Added 24 answers

Hint
If f is increasing, then f 1 is increasing.
Indeed, let u v and x,y such that u = f ( x ) and v = f ( y ). Then, since f is increasing,
u v f ( x ) f ( y ) x y f 1 ( u ) f 1 ( v ) .
PS: Use arcsin instead of sin 1
Janet Forbes

Janet Forbes

Expert

2022-07-12Added 4 answers

HINT:
d d x ( sin 1 x ) = 1 1 x 2 > 0
        1 < x < 1

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