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](https://www.expertsmind.com/CMSImages/706_inverse%20cosec.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.