Inverse trigonometric functions: Difference between revisions

Revision as of 15:00, 25 August 2025

Latest revision as of 16:43, 25 August 2025

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===<span class="anchor" id="principal_value_anchor">Principal values</span>===

Since none of the six trigonometric functions are not [[One-to-one function|one-to-one]], they must be restricted in order to have inverse functions. Therefore, the result [[Range of a function|range]]s of the inverse functions are proper (i.e. strict) [[subset]]s of the domains of the original functions.

For example, using {{em|function}} in the sense of [[multivalued function]]s, just as the [[square root]] function <math>y = \sqrt{x}</math> could be defined from <math>y^2 = x,</math> the function <math>y = \arcsin(x)</math> is defined so that <math>\sin(y) = x.</math> For a given real number <math>x,</math> with <math>-1 \leq x \leq 1,</math> there are multiple (in fact, [[countably infinite]]ly many) numbers <math>y</math> such that <math>\sin(y) = x</math>; for example, <math>\sin(0) = 0,</math> but also <math>\sin(\pi) = 0,</math> <math>\sin(2 \pi) = 0,</math> etc. When only one value is desired, the function may be restricted to its [[principal branch]]. With this restriction, for each <math>x</math> in the ___domain, the expression <math>\arcsin(x)</math> will evaluate only to a single value, called its [[principal value]]. These properties apply to all the inverse trigonometric functions.