Relationship between inverse and sine function, Mathematics

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Relationship between the inverse sine function and the sine function

We have the given relationship among the inverse sine function and the sine function.

                   sin (sin -1 x ) = x                                     sin -1 (sin x ) =x

In other terms they are inverses of each other. it means that we can utilizes the fact above to determine the derivative of inverse sine.  Let's start with,

f ( x ) = sin x                                                g ( x )= sin -1 x

Then,

g′ ( x ) = 1/ f ′ ( g ( x )) =         1 /cos (sin -1  x )

It is not an extremely useful formula.  Let's see if we can obtain better formula.  Let's begin by recalling the definition of inverse sine function.

                                          y ( sin -1 ( x )         ⇒ x = sin ( y )

By using the first part of this definition the denominator into the derivative becomes,

                                                 cos (sin -1 x ) = cos ( y )

Now, recall that

cos2  y + sin 2  y =1      ⇒  1769_inverse trig.png

By using this, now the denominator is,

547_inverse trig1.png

Now, utilizes the second part of the definition of the inverse sine function. Then the denominator is,

210_inverse trig 2.png

Putting all this together gives the given derivative.

2082_inverse trig3.png


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