Inverse cosine, Mathematics

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Inverse Cosine : Now see at inverse cosine.  Following is the definition for the inverse cosine.

                        y = cos-1 x       ⇔ cos y = x                   for     0 ≤ y ≤ ?

As with the inverse since we've got a restriction on the angles, y, which we get out of the inverse cosine function. Again, if you'd like to verify it a quick sketch of unit circle should convince you that this range will cover all possible values of cosine exactly once.  Also, we have

                          -1 ≤ x ≤ 1 because -1 ≤ cos ( y ) ≤ 1.

Example   Evaluate cos-1   (-√2/  2)

Solution : As with the inverse sine we are actually just asking the following.

                                                       cos y = - √2 /2

where y have to meet the requirements given above.  From a unit circle we can illustrates that we must have y =3 ∏/4    .

The inverse cosine & cosine functions are also inverses of each other and therefore we have,

cos (cos-1  x ) = x                                          cos-1 (cos x ) =x

To determine the derivative we'll do the similar kind of work which we did with the inverse sine above.  If we begin with then,

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

then

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

Here Simplifying the denominator is almost alike to the work we did for the inverse sine & so isn't illustrated here.  Upon simplifying we get the given derivative.

706_inverse cosec.png

Therefore, the derivative of the inverse cosine is closely identical to the derivative of the inverse sine. The single difference is the negative sign.


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