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arcsine: the inverse function of sine (usually taken on the interval
by default).
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arccosine: the inverse function of cosine (usually taken on the interval
by default).
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arctangent: the inverse function of tangent (usually taken on the
interval by default).
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Any of these inverse functions could be defined on intervals other than
those given above. There are an infinite number of inverses of sine, for
example, depending on which interval we choose (sine simply needs to be
one-to-one on that interval).
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Exercise 17: we want to find the derivative of
, which we'll
do using implicit differentiation:
The same method of implicit differentiation is demonstrated in the text, to find the derivatives of arcsine and arctan.
The trigonometric functions are obviously very important functions, and their inverses are likewise useful. (They are not, however, in the same class as the exponential and logarithmic functions; at least not in my experience. And I've never had need for the other arc functions.)
All these functions do appear as antiderivatives of certain classes of functions, as you can see from the derivatives of these functions. Spend a little time getting to know them.