Editing Inverse trigonometric functions (section)

==Basic concepts==
[[File:TrigFunctionDiagram.svg|thumb|The points labelled {{color|#D00|1}}, {{color|#02D|Sec(''θ'')}}, {{color|#0D1|Csc(''θ'')}} represent the length of the line segment from the origin to that point. {{color|#D00|Sin(''θ'')}}, {{color|#02D|Tan(''θ'')}}, and {{color|#0D1|1}} are the heights to the line starting from the {{mvar|x}}-axis, while {{color|#D00|Cos(''θ'')}}, {{color|#02D|1}}, and {{color|#0D1|Cot(''θ'')}} are lengths along the {{mvar|x}}-axis starting from the origin.]]

===<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
}}

=== Solutions to elementary trigonometric equations ===

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of <math>2 \pi:</math>

* Sine and cosecant begin their period at <math display="inline">2 \pi k-\frac{\pi}{2}</math> (where <math>k</math> is an integer), finish it at <math display="inline">2 \pi k+\frac{\pi}{2},</math> and then reverse themselves over <math display="inline">2 \pi k+\frac{\pi}{2}</math> to <math display="inline">2 \pi k+\frac{3\pi}{2}.</math>
* Cosine and secant begin their period at <math>2 \pi k,</math> finish it at <math>2 \pi k+\pi.</math> and then reverse themselves over <math>2 \pi k+\pi</math> to <math>2 \pi k+2 \pi.</math>
* Tangent begins its period at <math display="inline">2 \pi k-\frac{\pi}{2},</math>  finishes it at <math display="inline">2 \pi k+\frac{\pi}{2},</math> and then repeats it (forward) over <math display="inline">2 \pi k+\frac{\pi}{2}</math> to <math display="inline">2 \pi k+\frac{3 \pi}{2}.</math> 
* Cotangent begins its period at <math>2 \pi k, </math> finishes it at <math>2 \pi k+\pi,</math> and then repeats it (forward) over <math>2 \pi k+\pi</math> to <math>2 \pi k+2 \pi.</math>

This periodicity is reflected in the general inverses, where <math>k</math> is some integer.

The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions. 
It is assumed that the given values <math>\theta,</math> <math>r,</math> <math>s,</math> <math>x,</math> and <math>y</math> all lie within appropriate ranges so that the relevant expressions below are [[well-defined]]. 
Note that "for some <math>k \in \Z</math>" is just another way of saying "for some [[integer]] <math>k.</math>"

The symbol <math>\,\iff\,</math> is [[logical equality]] and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote<ref group="note">The expression "LHS <math>\,\iff\,</math> RHS" indicates that {{em|either}} (a) the left hand side (i.e. LHS) and right hand side (i.e. RHS) are {{em|both}} true, or else (b) the left hand side and right hand side are {{em|both}} false; there is {{em|no}} option (c) (e.g. it is {{em|not}} possible for the LHS statement to be true and also simultaneously for the RHS statement to be false), because otherwise "LHS <math>\,\iff\,</math> RHS" would not have been written. <br/> 

To clarify, suppose that it is written "LHS <math>\,\iff\,</math> RHS" where LHS (which abbreviates ''left hand side'') and RHS are both statements that can individually be either be true or false. For example, if <math>\theta</math> and <math>s</math> are some given and fixed numbers and if the following is written:
<math displaystyle="block">\tan \theta = s \,\iff\, \theta = \arctan(s)+\pi k \quad \text{ for some } k \in \Z</math>
then LHS is the statement "<math>\tan \theta = s</math>". Depending on what specific values <math>\theta</math> and <math>s</math> have, this LHS statement can either be true or false. For instance, LHS is true if <math>\theta = 0</math> and <math>s = 0</math> (because in this case <math>\tan \theta = \tan 0 = s</math>) but LHS is false if <math>\theta = 0</math> and <math>s = 2</math> (because in this case <math>\tan \theta = \tan 0 = s</math> which is not equal to <math>s = 2</math>); more generally, LHS is false if <math>\theta = 0</math> and <math>s \neq 0.</math> Similarly, RHS is the statement "<math>\theta = \arctan(s)+\pi k</math> for some <math>k \in \Z</math>". The RHS statement can also either true or false (as before, whether the RHS statement is true or false depends on what specific values <math>\theta</math> and <math>s</math> have). The logical equality symbol <math>\,\iff\,</math> means that (a) if the LHS statement is true then the RHS statement is also {{em|necessarily}} true, and moreover (b) if the LHS statement is false then the RHS statement is also {{em|necessarily}} false. Similarly, <math>\,\iff\,</math> {{em|also}} means that (c) if the RHS statement is true then the LHS statement is also {{em|necessarily}} true, and moreover (d) if the RHS statement is false then the LHS statement is also {{em|necessarily}} false.</ref> for more details and an example illustrating this concept).

{| class="wikitable" style="border: none;"
|+ 
|-
! Equation !! [[if and only if]] !! colspan="7" | Solution 
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\sin \theta = y</math>
| style="text-align: center;" |[[Logical equality|<math>\iff</math>]]
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta =\, </math>
| style='border-style: solid none solid none; text-align: right;' |<math>(-1)^k</math>
| style='border-style: solid none solid none; text-align: left;' |<math>\arcsin (y)</math>
| style='border-style: solid none solid none;' |<math>+</math>
| style='border-style: solid none solid none;' |
| style='border-style: solid none solid none; padding-right: 2em;' |<math>\pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END:   sin ''θ'' = ''x'' --------------->
|-
<!--------------- START: csc ''θ'' = ''r'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\csc \theta = r</math>
| style="text-align: center;" |<math>\iff</math>
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta =\, </math>
| style='border-style: solid none solid none; text-align: right;' |<math>(-1)^k</math>
| style='border-style: solid none solid none; text-align: left;' |<math>\arccsc (r)</math>
| style='border-style: solid none solid none;' |<math>+</math>
| style='border-style: solid none solid none;' |
| style='border-style: solid none solid none; padding-right: 2em;' |<math>\pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END:   csc ''θ'' = ''r'' --------------->
|-
<!--------------- START: cos ''θ'' = ''x'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\cos \theta = x</math>
| style="text-align: center;" |<math>\iff</math>
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta =\, </math>
| style='border-style: solid none solid none; text-align: right;' |<math>\pm\,</math>
| style='border-style: solid none solid none; text-align: left;' |<math>\arccos(x)</math>
| style='border-style: solid none solid none;' |<math>+</math>
| style='border-style: solid none solid none;' |<math>2</math>
| style='border-style: solid none solid none; padding-right: 2em;' |<math>\pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END:   cos ''θ'' = ''x'' --------------->
|-
<!--------------- START: sec ''θ'' = ''r'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\sec \theta = r</math>
| style="text-align: center;" |<math>\iff</math>
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta =\, </math>
| style='border-style: solid none solid none; text-align: right;' |<math>\pm\,</math>
| style='border-style: solid none solid none; text-align: left;' |<math>\arcsec (r)</math>
| style='border-style: solid none solid none;' |<math>+</math>
| style='border-style: solid none solid none;' |<math>2</math>
| style='border-style: solid none solid none; padding-right: 2em;' |<math>\pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
|-
<!--------------- END:   sec ''θ'' = ''r'' --------------->
|-
<!--------------- START: tan ''θ'' = ''s'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\tan \theta = s</math>
| style="text-align: center;" |<math>\iff</math>
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta =\, </math>
| style='border-style: solid none solid none; text-align: right;' |
| style='border-style: solid none solid none; text-align: left;' |<math>\arctan (s)</math>
| style='border-style: solid none solid none;' |<math>+</math>
| style='border-style: solid none solid none;' |
| style='border-style: solid none solid none; padding-right: 2em;' |<math>\pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END:   tan ''θ'' = ''s'' --------------->
|-
<!--------------- START: cot ''θ'' = ''r'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\cot \theta = r</math>
| style="text-align: center;" |<math>\iff</math>
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta =\, </math>
| style='border-style: solid none solid none; text-align: right;' |
| style='border-style: solid none solid none; text-align: left;' |<math>\arccot (r)</math>
| style='border-style: solid none solid none;' |<math>+</math>
| style='border-style: solid none solid none;' |
| style='border-style: solid none solid none; padding-right: 2em;' |<math>\pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END:   cot ''θ'' = ''r'' --------------->
|}

where the first four solutions can be written in expanded form as:

{| class="wikitable" style="border: none;"
|+ 
|-
! Equation !! [[if and only if]] !! colspan="7" | Solution 
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\sin \theta = y</math>
| style="text-align: center;" |[[Logical equality|<math>\iff</math>]]
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta = \;\;\;\,\arcsin(y)+2 \pi k</math> <br/>{{space|10}}or <br/><math>\theta =-\arcsin(y)+2 \pi k+\pi</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END:   sin ''θ'' = ''x'' --------------->
|-
<!--------------- START: csc ''θ'' = ''r'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\csc \theta = r</math>
| style="text-align: center;" |<math>\iff</math>
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta = \;\;\;\,\arccsc(r)+2 \pi k</math> <br/>{{space|10}}or <br/><math>\theta =-\arccsc(r)+2 \pi k+\pi</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END:   csc ''θ'' = ''r'' --------------->
|-
<!--------------- START: cos ''θ'' = ''x'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\cos \theta = x</math>
| style="text-align: center;" |<math>\iff</math>
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta = \;\;\;\,\arccos(x)+2 \pi k</math> <br/>{{space|9}}or <br/><math>\theta =-\arccos(x)+2 \pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END:   cos ''θ'' = ''x'' --------------->
|-
<!--------------- START: sec ''θ'' = ''r'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\sec \theta = r</math>
| style="text-align: center;" |<math>\iff</math>
| style='border-style: solid none solid none; text-align: left; padding-left: 2em;' |<math>\theta = \;\;\;\,\arcsec(r)+2 \pi k</math> <br/>{{space|9}}or <br/><math>\theta =-\arcsec(r)+2 \pi k</math>
| style="text-align: center; padding-left: 1em; padding-right: 1em;" |for some <math>k \in \Z</math>
<!--------------- END:   sec ''θ'' = ''r'' --------------->
|}

For example, if <math>\cos \theta = -1</math> then <math>\theta = \pi+2 \pi k = -\pi+2 \pi (1+k)</math> for some <math>k \in \Z.</math> While if <math>\sin \theta = \pm 1</math> then <math display="inline">\theta = \frac{\pi}{2}+\pi k =-\frac{\pi}{2}+\pi (k+1)</math> for some <math>k \in \Z,</math> where <math>k</math> will be even if <math>\sin \theta = 1</math> and it will be odd if <math>\sin \theta = -1.</math> The equations <math>\sec \theta = -1</math> and <math>\csc \theta = \pm 1</math> have the same solutions as <math>\cos \theta = -1</math> and <math>\sin \theta = \pm 1,</math> respectively. In all equations above {{em|except}} for those just solved (i.e. except for <math>\sin</math>/<math>\csc \theta = \pm 1</math> and <math>\cos</math>/<math>\sec \theta =-1</math>), the integer <math>k</math> in the solution's formula is uniquely determined by <math>\theta</math> (for fixed <math>r, s, x,</math> and <math>y</math>). 

With the help of [[Parity (mathematics)|integer parity]] 
<math display=block>\operatorname{Parity}(h) =
\begin{cases}
0 & \text{if } h \text{ is even } \\
1 & \text{if } h \text{ is odd } \\
\end{cases}</math>
it is possible to write a solution to <math>\cos \theta = x</math> that doesn't involve the "plus or minus" <math>\,\pm\,</math> symbol:  
:<math>cos \; \theta = x \quad</math> if and only if <math>\quad \theta = (-1)^h \arccos(x) + \pi h + \pi \operatorname{Parity}(h) \quad</math> for some <math>h \in \Z.</math> 
And similarly for the secant function,
:<math>sec \; \theta = r \quad</math> if and only if <math>\quad \theta = (-1)^h \arcsec(r) + \pi h + \pi \operatorname{Parity}(h) \quad</math> for some <math>h \in \Z,</math>
where <math>\pi h + \pi \operatorname{Parity}(h)</math> equals <math>\pi h</math> when the integer <math>h</math> is even, and equals <math>\pi h + \pi</math> when it's odd.

====Detailed example and explanation of the "plus or minus" symbol {{math|±}} ====

The solutions to <math>\cos \theta = x</math> and <math>\sec \theta = x</math> involve the "plus or minus" symbol <math>\,\pm,\,</math> whose meaning is now clarified. Only the solution to <math>\cos \theta = x</math> will be discussed since the discussion for <math>\sec \theta = x</math> is the same. 
We are given <math>x</math> between <math>-1 \leq x \leq 1</math> and we know that there is an angle <math>\theta</math> in some interval that satisfies <math>\cos \theta = x.</math> We want to find this <math>\theta.</math> The table above indicates that the solution is 
<math display="block">\,\theta = \pm \arccos x+2 \pi k\, \quad \text{ for some }k \in \Z</math>
which is a shorthand way of saying that (at least) one of the following statement is true: <br />
#<math>\,\theta = \arccos x+2 \pi k\,</math> for some integer <math>k,</math> <br/>or 
#<math>\,\theta =-\arccos x+2 \pi k\,</math> for some integer <math>k.</math> 
As mentioned above, if <math>\,\arccos x = \pi\,</math> (which by definition only happens when <math>x = \cos \pi = -1</math>) then both statements (1) and (2) hold, although with different values for the integer <math>k</math>: if <math>K</math> is the integer from statement (1), meaning that <math>\theta = \pi+2 \pi K</math> holds, then the integer <math>k</math> for statement (2) is <math>K+1</math> (because <math>\theta = -\pi+2 \pi (1+K)</math>). 
However, if <math>x \neq -1</math> then the integer <math>k</math> is unique and completely determined by <math>\theta.</math> 
If <math>\,\arccos x = 0\,</math>  (which by definition only happens when <math>x = \cos 0 = 1</math>) then <math>\,\pm\arccos x = 0\,</math> (because <math>\,+ \arccos x = +0 = 0\,</math> and <math>\,-\arccos x = -0 = 0\,</math> so in both cases <math>\,\pm \arccos x\,</math> is equal to <math>0</math>) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold). 
Having considered the cases <math>\,\arccos x = 0\,</math> and <math>\,\arccos x = \pi,\,</math> we now focus on the case where <math>\,\arccos x \neq 0\,</math> and <math>\,\arccos x \neq \pi,\,</math> So assume this from now on. The solution to <math>\cos \theta = x</math> is still
<math display="block">\,\theta = \pm \arccos x+2 \pi k\, \quad \text{ for some }k \in \Z</math>
which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because <math>\,\arccos x \neq 0\,</math> and <math>\,0 < \arccos x < \pi,\,</math> statements (1) and (2) are different and furthermore, ''exactly one'' of the two equalities holds (not both). Additional information about <math>\theta</math> is needed to determine which one holds. For example, suppose that <math>x = 0</math> and that {{em|all}} that is known about <math>\theta</math> is that <math>\,-\pi \leq \theta \leq \pi\,</math> (and nothing more is known). Then 
<math display="block">\arccos x = \arccos 0 = \frac{\pi}{2}</math> 
and moreover, in this particular case <math>k = 0</math> (for both the <math>\,+\,</math> case and the <math>\,-\,</math> case) and so consequently,
<math display="block">\theta ~=~ \pm \arccos x+2 \pi k ~=~ \pm \left(\frac{\pi}{2}\right)+2\pi (0) ~=~ \pm \frac{\pi}{2}.</math>
This means that <math>\theta</math> could be either <math>\,\pi/2\,</math> or <math>\,-\pi/2.</math> Without additional information it is not possible to determine which of these values <math>\theta</math> has. 
An example of some additional information that could determine the value of <math>\theta</math> would be knowing that the angle is above the <math>x</math>-axis (in which case <math>\theta = \pi/2</math>) or alternatively, knowing that it is below the <math>x</math>-axis (in which case <math>\theta =-\pi/2</math>). 

==== Equal identical trigonometric functions ====

{{EqualOrNegativeIdenticalTrigonometricFunctionsSolutions|includeTableDescription=true|style=}}

;Set of all solutions to elementary trigonometric equations

Thus given a single solution <math>\theta</math> to an elementary trigonometric equation (<math>\sin \theta = y</math> is such an equation, for instance, and because <math>\sin (\arcsin y) = y</math> always holds, <math>\theta := \arcsin y</math> is always a solution), the set of all solutions to it are:

{| class="wikitable" style="border: none;"
|+ 
|-
! If <math>\theta</math> solves !! then !! colspan="7" | Set of all solutions (in terms of <math>\theta</math>)
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\sin \theta = y</math>
| style="text-align: center;" |then
| style='border-style: solid none solid none; text-align: left; padding-left: 1em;' |<math>\{\varphi:\sin \varphi=y\} =\, </math>
| style='border-style: solid none solid none; text-align: right; padding: 0;' |<math>(\theta</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\,+\, 2</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\pi \Z)</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\,\cup\, (-\theta</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>-\pi</math>
| style='border-style: solid solid solid none; text-align: left;  padding-left: 0; padding-right: 2em;' |<math>+ 2 \pi \Z)</math>
<!--------------- END:   sin ''θ'' = ''x'' --------------->
|-
<!--------------- START: csc ''θ'' = ''r'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\csc \theta = r</math>
| style="text-align: center;" |then
| style='border-style: solid none solid none; text-align: left; padding-left: 1em;' |<math>\{\varphi:\csc \varphi=r\} =\, </math>
| style='border-style: solid none solid none; text-align: right; padding: 0;' |<math>(\theta</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\,+\, 2</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\pi \Z)</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\,\cup\, (-\theta</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>-\pi</math>
| style='border-style: solid solid solid none; text-align: left;  padding-left: 0; padding-right: 2em;' |<math>+ 2 \pi \Z)</math>
<!--------------- END:   csc ''θ'' = ''r'' --------------->
|-
<!--------------- START: cos ''θ'' = ''x'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\cos \theta = x</math>
| style="text-align: center;" |then
| style='border-style: solid none solid none; text-align: left; padding-left: 1em;' |<math>\{\varphi:\cos \varphi=x\} =\, </math>
| style='border-style: solid none solid none; text-align: right; padding: 0;' |<math>(\theta</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\,+\, 2</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\pi \Z)</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\,\cup\, (-\theta</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |
| style='border-style: solid solid solid none; text-align: left;  padding-left: 0; padding-right: 2em;' |<math>+ 2 \pi \Z)</math>
<!--------------- END:   cos ''θ'' = ''x'' --------------->
|-
<!--------------- START: sec ''θ'' = ''r'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\sec \theta = r</math>
| style="text-align: center;" |then
| style='border-style: solid none solid none; text-align: left; padding-left: 1em;' |<math>\{\varphi:\sec \varphi=r\} =\, </math>
| style='border-style: solid none solid none; text-align: right; padding: 0;' |<math>(\theta</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\,+\, 2</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\pi \Z)</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\,\cup\, (-\theta</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |
| style='border-style: solid solid solid none; text-align: left;  padding-left: 0; padding-right: 2em;' |<math>+ 2 \pi \Z)</math>
|-
<!--------------- END:   sec ''θ'' = ''r'' --------------->
|-
<!--------------- START: tan ''θ'' = ''s'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\tan \theta = s</math>
| style="text-align: center;" |then
| style='border-style: solid none solid none; text-align: left; padding-left: 1em;' |<math>\{\varphi:\tan \varphi=s\} =\, </math>
| style='border-style: solid none solid none; text-align: right; padding: 0;' |<math>\theta</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\,+\,</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\pi \Z</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |
| style='border-style: solid solid solid none; text-align: left;  padding-left: 0; padding-right: 2em;' |
<!--------------- END:   tan ''θ'' = ''s'' --------------->
|-
<!--------------- START: cot ''θ'' = ''r'' --------------->
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\cot \theta = r</math>
| style="text-align: center;" |then
| style='border-style: solid none solid none; text-align: left; padding-left: 1em;' |<math>\{\varphi:\cot \varphi=r\} =\, </math>
| style='border-style: solid none solid none; text-align: right; padding: 0;' |<math>\theta</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\,+\,</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |<math>\pi \Z</math>
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |
| style='border-style: solid none solid none; text-align: left;  padding: 0;' |
| style='border-style: solid solid solid none; text-align: left;  padding-left: 0; padding-right: 2em;' |
<!--------------- END:   cot ''θ'' = ''r'' --------------->
|}

===Transforming equations===

The equations above can be transformed by using the reflection and shift identities:<ref>{{harvnb|Abramowitz|Stegun|1972|loc=p. 73, 4.3.44}}</ref>

{| class="wikitable" style="text-align: center;"
|+ Transforming equations by shifts and reflections
|-
! scope="col" | Argument: <math>\underline{\;~~~~~~\;}=</math>
! scope="col" |<math>-\theta</math>
! scope="col" |<math>\frac{\pi}{2} \pm \theta</math>
! scope="col" |<math>\pi \pm \theta</math>
! scope="col" |<math>\frac{3\pi}{2} \pm \theta</math>
! scope="col" |<math>2 k \pi \pm \theta,</math><br/> <math>(k \in \Z)</math>
|- <!-- sin -->
! scope="row" |<math>\sin \underline{\;~~~~~~~~~~~~~~\;}=</math>
| <math>-\sin \theta</math>
| <math>\phantom{-}\cos \theta</math>
| <math>\mp\sin \theta</math>
| <math>-\cos \theta</math>
| <math>\pm\sin \theta</math>
|- <!-- csc -->
! scope="row" |<math>\csc \underline{\;~~~~~~~~~~~~~~\;}=</math>
| <math>-\csc \theta</math>
| <math>\phantom{-}\sec \theta</math>
| <math>\mp\csc \theta</math>
| <math>-\sec \theta</math>
| <math>\pm\csc \theta</math>
|- <!-- cos -->
! scope="row" |<math>\cos \underline{\;~~~~~~~~~~~~~~\;}=</math>
| <math>\phantom{-}\cos \theta</math>
| <math>\mp\sin \theta</math>
| <math>-\cos \theta</math>
| <math>\pm\sin \theta</math>
| <math>\phantom{-}\cos \theta</math>
|- <!-- sec -->
! scope="row" |<math>\sec \underline{\;~~~~~~~~~~~~~~\;}=</math>
| <math>\phantom{-}\sec \theta</math>
| <math>\mp\csc \theta</math>
| <math>-\sec \theta</math>
| <math>\pm\csc \theta</math>
| <math>\phantom{-}\sec \theta</math>
|- <!-- tan -->
! scope="row" |<math>\tan \underline{\;~~~~~~~~~~~~~~\;}=</math>
| <math>-\tan \theta</math>
| <math>\mp\cot \theta</math>
| <math>\pm\tan \theta</math>
| <math>\mp\cot \theta</math>
| <math>\pm\tan \theta</math>
|- <!-- cot -->
! scope="row" |<math>\cot \underline{\;~~~~~~~~~~~~~~\;}=</math>
| <math>-\cot \theta</math>
| <math>\mp\tan \theta</math>
| <math>\pm\cot \theta</math>
| <math>\mp\tan \theta</math>
| <math>\pm\cot \theta</math>
|}

These formulas imply, in particular, that the following hold:

<math display=block>
\begin{align}
\sin \theta
&= -\sin(-\theta)
&&= -\sin(\pi+\theta)
&&= \phantom{-}\sin(\pi-\theta) \\
&= -\cos\left(\frac{\pi}{2}+\theta\right)
&&= \phantom{-}\cos\left(\frac{\pi}{2}-\theta\right)
&&= -\cos\left(-\frac{\pi}{2}-\theta\right) \\
&= \phantom{-}\cos\left(-\frac{\pi}{2}+\theta\right)
&&= -\cos\left(\frac{3\pi}{2}-\theta\right)
&&= -\cos\left(-\frac{3\pi}{2}+\theta\right)
\\[0.3ex]
\cos \theta
&= \phantom{-}\cos(-\theta)
&&= -\cos(\pi+\theta)
&&= -\cos(\pi-\theta) \\
&= \phantom{-}\sin\left(\frac{\pi}{2}+\theta\right)
&&= \phantom{-}\sin\left(\frac{\pi}{2}-\theta\right) 
&&= -\sin\left(-\frac{\pi}{2}-\theta\right) \\
&= -\sin\left(-\frac{\pi}{2}+\theta\right)
&&= -\sin\left(\frac{3\pi}{2}-\theta\right)
&&= \phantom{-}\sin\left(-\frac{3\pi}{2}+\theta\right)
\\[0.3ex]
\tan \theta
&= -\tan(-\theta)
&&= \phantom{-}\tan(\pi+\theta)
&&= -\tan(\pi-\theta) \\
&= -\cot\left(\frac{\pi}{2}+\theta\right)
&&= \phantom{-}\cot\left(\frac{\pi}{2}-\theta\right)
&&= \phantom{-}\cot\left(-\frac{\pi}{2}-\theta\right) \\
&= -\cot\left(-\frac{\pi}{2}+\theta\right)
&&= \phantom{-}\cot\left(\frac{3\pi}{2}-\theta\right)
&&= -\cot\left(-\frac{3\pi}{2}+\theta\right)
\\[0.3ex]
\end{align}
</math>

where swapping <math>\sin \leftrightarrow \csc,</math> swapping <math>\cos \leftrightarrow \sec,</math> and swapping <math>\tan \leftrightarrow \cot</math> gives the analogous equations for <math>\csc, \sec, \text{ and } \cot,</math> respectively.

So for example, by using the equality <math display="inline">\sin \left(\frac{\pi}{2}-\theta\right) = \cos \theta,</math> the equation <math>\cos \theta = x</math> can be transformed into <math display="inline">\sin \left(\frac{\pi}{2}-\theta\right) = x,</math> which allows for the solution to the equation <math>\;\sin \varphi = x\;</math> (where <math display="inline">\varphi := \frac{\pi}{2}-\theta</math>) to be used; that solution being: 
<math>\varphi  = (-1)^k \arcsin (x)+\pi k \; \text{ for some } k \in \Z,</math>
which becomes:
<math display="block">\frac{\pi}{2}-\theta ~=~ (-1)^k \arcsin (x)+\pi k \quad \text{ for some } k \in \Z</math>
where using the fact that <math>(-1)^{k} = (-1)^{-k}</math> and substituting <math>h :=-k</math> proves that another solution to <math>\;\cos \theta = x\;</math> is:
<math display="block">\theta ~=~ (-1)^{h+1} \arcsin (x)+\pi h+\frac{\pi}{2} \quad \text{ for some } h \in \Z.</math>
The substitution <math>\;\arcsin x = \frac{\pi}{2}-\arccos x\;</math> may be used express the right hand side of the above formula in terms of <math>\;\arccos x\;</math> instead of <math>\;\arcsin x.\;</math>

=== Relationships between trigonometric functions and inverse trigonometric functions ===

Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length <math>x,</math> then applying the [[Pythagorean theorem]] and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that <math>x</math> is positive, and thus the result has to be corrected through the use of [[absolute value]]s and the [[Sign function|signum]] (sgn) operation.

{|class="wikitable"
|-
!<math>\theta</math>
!<math>\sin(\theta)</math>
!<math>\cos(\theta)</math>
!<math>\tan(\theta)</math>
!Diagram
|-
!<math>\arcsin(x)</math>
|<math>\sin(\arcsin(x)) = x </math>
|<math>\cos(\arcsin(x)) = \sqrt{1-x^2}</math>
|<math>\tan(\arcsin(x)) = \frac{x}{\sqrt{1-x^2}}</math>
|[[File:Trigonometric functions and inverse3.svg|150px]]
|-
!<math>\arccos(x)</math>
|<math>\sin(\arccos(x)) = \sqrt{1-x^2}</math>
|<math>\cos(\arccos(x)) = x </math>
|<math>\tan(\arccos(x)) = \frac{\sqrt{1-x^2}}{x}</math>
|[[File:Trigonometric functions and inverse.svg|150px]]
|-
!<math>\arctan(x)</math>
|<math>\sin(\arctan(x)) = \frac{x}{\sqrt{1+x^2}}</math>
|<math>\cos(\arctan(x)) = \frac{1}{\sqrt{1+x^2}}</math>
|<math>\tan(\arctan(x)) = x</math>
|[[File:Trigonometric functions and inverse2.svg|150px]]
|-
!<math>\arccot(x)</math>
|<math>\sin(\arccot(x)) = \frac{1}{\sqrt{1+x^2}}</math>
|<math>\cos(\arccot(x)) = \frac{x}{\sqrt{1+x^2}}</math>
|<math>\tan(\arccot(x)) = \frac{1}{x}</math>
|[[File:Trigonometric functions and inverse4.svg|150px]]
|-
!<math>\arcsec(x)</math>
|<math>\sin(\arcsec(x)) = \frac{\sqrt{x^2-1}}{|x|}</math>
|<math>\cos(\arcsec(x)) = \frac{1}{x}</math>
|<math>\tan(\arcsec(x)) = \sgn(x)\sqrt{x^2-1}</math>
|[[File:Trigonometric functions and inverse6.svg|150px]]
|-
!<math>\arccsc(x)</math>
|<math>\sin(\arccsc(x)) = \frac{1}{x}</math>
|<math>\cos(\arccsc(x)) = \frac{\sqrt{x^2-1}}{|x|}</math>
|<math>\tan(\arccsc(x)) = \frac{\sgn(x)}{\sqrt{x^2-1}}</math>
|[[File:Trigonometric functions and inverse5.svg|150px]]
|-
|}

===Relationships among the inverse trigonometric functions===
[[Image:Arcsine Arccosine.svg|168px|right|thumb|The usual principal values of the arcsin(''x'') (red) and arccos(''x'') (blue) functions graphed on the cartesian plane.]]
[[Image:Arctangent Arccotangent.svg|294px|right|thumb|The usual principal values of the arctan(''x'') and arccot(''x'') functions graphed on the cartesian plane.]]
[[Image:Arcsecant Arccosecant.svg|294px|right|thumb|Principal values of the arcsec(''x'') and arccsc(''x'') functions graphed on the cartesian plane.]]

Complementary angles:
:<math>\begin{align}
\arccos(x) &= \frac{\pi}{2} - \arcsin(x) \\[0.5em]
\arccot(x) &= \frac{\pi}{2} - \arctan(x) \\[0.5em]
\arccsc(x) &= \frac{\pi}{2} - \arcsec(x)
\end{align}</math>

Negative arguments:
:<math>\begin{align}
\arcsin(-x) &= -\arcsin(x) \\
\arccsc(-x) &= -\arccsc(x) \\
\arccos(-x) &= \pi -\arccos(x) \\
\arcsec(-x) &= \pi -\arcsec(x) \\
\arctan(-x) &= -\arctan(x) \\
\arccot(-x) &= \pi -\arccot(x)
\end{align}</math>

Reciprocal arguments:
:<math>\begin{align}
\arcsin\left(\frac{1}{x}\right) &= \arccsc(x) & \\[0.3em]
\arccsc\left(\frac{1}{x}\right) &= \arcsin(x) & \\[0.3em]
\arccos\left(\frac{1}{x}\right) &= \arcsec(x) & \\[0.3em]
\arcsec\left(\frac{1}{x}\right) &= \arccos(x) & \\[0.3em]
\arctan\left(\frac{1}{x}\right) &= \arccot(x)       &= \frac{\pi}{2} - \arctan(x) \, , \text{ if } x > 0 \\[0.3em]
\arctan\left(\frac{1}{x}\right) &= \arccot(x) - \pi &= -\frac{\pi}{2} - \arctan(x) \, , \text{ if } x < 0 \\[0.3em]
\arccot\left(\frac{1}{x}\right) &= \arctan(x)       &= \frac{\pi}{2} - \arccot(x) \, , \text{ if } x > 0 \\[0.3em]
\arccot\left(\frac{1}{x}\right) &= \arctan(x) + \pi &= \frac{3\pi}{2} - \arccot(x) \, , \text{ if } x < 0 
\end{align}</math>
The identities above can be used with (and derived from) the fact that <math>\sin</math> and <math>\csc</math> are [[Multiplicative inverse|reciprocals]] (i.e. <math>\csc = \tfrac1{\sin}</math>), as are <math>\cos</math> and <math>\sec,</math> and <math>\tan</math> and <math>\cot.</math>

Useful identities if one only has a fragment of a sine table:
:<math>\begin{align}
\arcsin(x) &= \frac{1}{2}\arccos\left(1-2x^2\right) \, , \text{ if } 0 \leq x \leq 1 \\
\arcsin(x) &= \arctan\left(\frac{x}{\sqrt{1 - x^2}}\right) \\
\arccos(x) &= \frac{1}{2}\arccos\left(2x^2-1\right) \, , \text{ if } 0 \leq x \leq 1 \\
\arccos(x) &= \arctan\left(\frac{\sqrt{1 - x^2}}{x}\right) \\
\arccos(x) &= \arcsin\left(\sqrt{1 - x^2}\right) \, , \text{ if } 0 \leq x \leq 1 \text{ , from which you get } \\
\arccos    &\left(\frac{1-x^2}{1 + x^2}\right) = \arcsin \left (\frac{2x}{1 + x^2}\right) \, , \text{ if } 0 \leq x \leq 1 \\
\arcsin    &\left(\sqrt{1 - x^2}\right) =\frac{\pi}{2}-\sgn(x)\arcsin(x) \\
\arctan(x) &= \arcsin\left(\frac{x}{\sqrt{1 + x^2}}\right) \\
\arccot(x) &= \arccos\left(\frac{x}{\sqrt{1 + x^2}}\right)
\end{align}</math>

Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).

A useful form that follows directly from the table above is

:<math>\arctan(x) = \arccos\left(\sqrt{\frac{1}{1+x^2}}\right)\, , \text{ if } x\geq 0 </math>.

It is obtained by recognizing that <math>\cos\left(\arctan\left(x\right)\right) = \sqrt{\frac{1}{1+x^2}} = \cos\left(\arccos\left(\sqrt{\frac{1}{1+x^2}}\right)\right)</math>.

From the [[tangent half-angle formula|half-angle formula]], <math>\tan\left(\tfrac{\theta}{2}\right) = \tfrac{\sin(\theta)}{1 + \cos(\theta)}</math>, we get:
:<math>\begin{align}
\arcsin(x) &= 2 \arctan\left(\frac{x}{1 + \sqrt{1 - x^2}}\right) \\[0.5em]
\arccos(x) &= 2 \arctan\left(\frac{\sqrt{1 - x^2}}{1 + x}\right) \, , \text{ if } -1 < x \leq  1 \\[0.5em]
\arctan(x) &= 2 \arctan\left(\frac{x}{1 + \sqrt{1 + x^2}}\right)
\end{align}</math>

===Arctangent addition formula===
:<math>\arctan(u) \pm \arctan(v) = \arctan\left(\frac{u \pm v}{1 \mp uv}\right) \pmod \pi \, , \quad u v \ne 1 \, .</math>
This is derived from the tangent [[Angle sum and difference identities|addition formula]]
:<math>\tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)} \, ,</math>
by letting
:<math>\alpha = \arctan(u) \, , \quad \beta = \arctan(v) \, .</math>