## Quick Reference

The inverse sine function sin^{−1} is, to put it briefly, the inverse function of sine, so that *y*=sin^{−1}*x* if *x*=sin *y*. Thus sin^{−1} ½ = *π*/6 because sin(*π*/6) = ½. However, sin(5*π*/6) = ½ also, so it might be thought that sin^{−1} ½ = *π*/6 or 5π/6. It is necessary to avoid such ambiguity so it is normally agreed that the value to be taken is the one lying in the interval [−π/2, π/2]. Similarly, *y*=tan^{−1}*x* if *x*=tan *y*, and the value *y* is taken to lie in the interval (−π/2, π/2). Also, *y*=cos^{−1}*x* if *x*=cos *y* and the value *y* is taken to lie in the interval [0,π].

A more advanced approach provides more explanation. The inverse function of a trigonometric function exists only if the original function is restricted to a suitable domain. This can be an interval *I* in which the function is strictly increasing or strictly decreasing (see inverse function). So, to obtain an inverse function for sin *x*, the function is restricted to a domain consisting of the interval [−π/2, π/2]; tan *x* is restricted to the interval (−π/2, π/2); and cos *x* is restricted to [0, π]. The domain of the inverse function is, in each case, the range of the restricted function. Hence the following inverse functions are obtained: sin^{−1}: [−1, 1] → [−π/2, π/2]; tan^{−1}: **R** → (−π/2, π/2); cos^{−1}[−1, 1] → [0, π]. The notation arcsin, arctan and arccos, for sin^{−1}, tan^{−1} and cos^{−1}, is also used. The following derivatives can be obtained:

*Subjects:*
Mathematics.

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