Editing Inverse trigonometric functions (section)

===<span class="anchor" id="principal_value_anchor">Principal values</span>===
Since none of the six trigonometric functions are [[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.

The principal inverses are listed in the following table.

{| class="wikitable" style="text-align:center"
|- 
! scope="col" | Name
! scope="col" | Usual notation
! scope="col" | Definition
! scope="col" | Domain of {{mvar|x}} for real result
! scope="col" | Range of usual principal value <br/>([[radian]]s)
! scope="col" | Range of usual principal value <br/>([[Degree (angle)|degrees]])
|-
! scope="row" | arcsine
| {{math|1= ''y'' = arcsin(''x'')}} || {{math|1=''x'' = [[sine|sin]](''y'')}} || {{math|−1 ≤ ''x'' ≤ 1}} || {{math|−{{sfrac|π|2}} ≤ ''y'' ≤ {{sfrac|π|2}}}} || {{math|−90° ≤ ''y'' ≤ 90°}}
|-
! scope="row" | arccosine
| {{math|1= ''y'' = arccos(''x'')}} || {{math|1=''x'' = [[cosine|cos]](''y'')}} || {{math|−1 ≤ ''x'' ≤ 1}} || {{math|0 ≤ ''y'' ≤ π}} || {{math|0° ≤ ''y'' ≤ 180°}}
|-
! scope="row" | arctangent
| {{math|1= ''y'' = arctan(''x'')}} || {{math|1=''x'' = [[Tangent (trigonometry)|tan]](''y'')}} || all real numbers || {{math|−{{sfrac|π|2}} < ''y'' < {{sfrac|π|2}}}} || {{math|−90° < ''y'' < 90°}}
|-
! scope="row" | arccotangent
| {{math|1= ''y'' = arccot(''x'')}} || {{math|1=''x'' = [[cotangent|cot]](''y'')}} || all real numbers || {{math|0 < ''y'' < π}} || {{math|0° < ''y'' < 180°}}
|-
! scope="row" | arcsecant
| {{math|1= ''y'' = arcsec(''x'')}} || {{math|1=''x'' = [[Secant (trigonometry)|sec]](''y'')}} || {{math|{{abs|''x''}} ≥ 1}} || {{math|0 ≤ ''y'' < {{sfrac|π|2}}}} or {{math|{{sfrac|π|2}} < ''y'' ≤ π}} || {{math|0° ≤ ''y'' < 90°}} or {{math|90° < ''y'' ≤ 180°}}
|-
! scope="row" | arccosecant
| {{math|1= ''y'' = arccsc(''x'')}} ||{{math|1=''x'' = [[cosecant|csc]](''y'')}} || {{math|{{abs|''x''}} ≥ 1}} || {{math|−{{sfrac|π|2}} ≤ ''y'' < 0}} or {{math|0 < ''y'' ≤ {{sfrac|π|2}}}} || {{math|−90° ≤ ''y'' < 0}} or {{math|0° < ''y'' ≤ 90°}}
|-
|}
Note: Some authors define the range of arcsecant to be {{nowrap|(<math display="inline">0 \leq y < \frac{\pi}{2}</math>}} or {{nowrap|<math display="inline">\pi \leq y < \frac{3 \pi}{2}</math> ),}}<ref>For example: {{pb}} {{cite book |last1=Stewart |first1=James |last2=Clegg |first2=Daniel |last3=Watson |first3=Saleem |year=2021 |title=Calculus: Early Transcendentals |edition=9th |isbn=978-1-337-61392-7 |chapter=Inverse Functions and Logarithms |publisher=Cengage Learning |at=§ 1.5, {{pgs|64}} }}</ref> because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, <math>\tan(\arcsec(x)) = \sqrt{x^2 - 1},</math> whereas with the range {{nowrap|(<math display="inline">0 \leq y < \frac{\pi}{2}</math>}} or {{nowrap|<math display="inline">\frac{\pi}{2} < y \leq \pi</math>),}} we would have to write <math>\tan(\arcsec(x)) = \pm \sqrt{x^2 - 1},</math> since tangent is nonnegative on <math display="inline">0 \leq y < \frac{\pi}{2},</math> but nonpositive on <math display="inline">\frac{\pi}{2} < y \leq \pi.</math> For a similar reason, the same authors define the range of arccosecant to be <math display="inline">( - \pi < y \leq - \frac{\pi}{2}</math> or <math display="inline">0 < y \leq \frac{\pi}{2} ) .</math>

====Domains====

If {{mvar|x}} is allowed to be a [[complex number]], then the range of {{mvar|y}} applies only to its real part.

{{DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions
 |includeTableDescription=true
 |includeExplanationOfNotation=true
}}