Arc function

Arc functions (from Latin arcus " arc "), also called cyclometric functions , are, as their alternative designation as inverse trigonometric functions suggests, inverse functions of trigonometric functions  - the arc functions therefore provide the corresponding angle for a given trigonometric function value .

There is an arc function for each of the six trigonometric functions, which is distinguished in mathematical formulas and equations by a preceding or abbreviation of the associated trigonometric function. Especially in English-speaking countries, but also on the keyboards of most pocket calculators, there is an increasing number of notations with the exponent −1, which is intended to signal that it is the inverse function (but not the reciprocal value ) of the said angle function: ${\displaystyle \operatorname {arc} }$ ${\displaystyle \operatorname {a} }$

Angle function	Arc function	Abbreviation	alternative abbreviation
Sine	Arcsine	${\displaystyle \arcsin }$ or ${\displaystyle \operatorname {asin} }$	${\displaystyle \sin ^{-1}}$
cosine	Arccosine	${\displaystyle \arccos }$ or ${\displaystyle \operatorname {acos} }$	${\displaystyle \cos ^{-1}}$
tangent	Arctangent	${\displaystyle \arctan }$ or ${\displaystyle \operatorname {atan} }$	${\displaystyle \tan ^{-1}}$
cotangent	Arccotangent	${\displaystyle \operatorname {arccot} }$ or ${\displaystyle \operatorname {acot} }$	${\displaystyle \cot ^{-1}}$
Secans	Arc secans	${\displaystyle \operatorname {arcsec} }$	${\displaystyle \sec ^{-1}}$
cosecant	Arccosecans	${\displaystyle \operatorname {arccsc} }$	${\displaystyle \csc ^{-1}}$

The main values ​​of the arcsin ( x ) (red) and arccos ( x ) functions (blue)	The main values ​​of the arctan ( x ) (red) and arccot ​​( x ) functions (blue)
	The main values ​​of the arcsec ( x ) and arccsc ( x ) functions

Riemann area of the complex logarithm . The leaves are spaced apart by . ${\displaystyle 2\pi }$

Since the trigonometric functions are periodic functions , they are initially not invertible . However, if you limit yourself to a monotony interval of the respective output function, z. B. to the interval or , the restricted function thus obtained can very well be inverted. However, the monotony intervals only cover half a period, see figure above. However, if you know both the sine and the cosine of an angle (more generally: complex components), you can determine the angle up to entire periods , see figure on the right for the illustration and atan2 for the calculation. ${\displaystyle [-\pi /2,\pi /2]}$ ${\displaystyle [0,\pi ]}$ ${\displaystyle (2\pi )}$

Relationships between functions

See also: Trigonometric function: relationships between functions

The arc functions can be converted into each other as follows:

	arcsin	arccos	arctan	arccot	arcsec	arccsc
arcsin (x)	${\displaystyle \,\arcsin(x)}$	${\displaystyle \,\arccos \left(\pm {\sqrt {1-x^{2}}}\right)}$	${\displaystyle \,\arctan \left(\pm {\frac {x}{\sqrt {1+x^{2}}}}\right)}$	${\displaystyle \,\operatorname {arccot} \left(\pm {\frac {1}{\sqrt {x^{2}+1}}}\right)}$	${\displaystyle \,\operatorname {arcsec} \left(\pm {\frac {\sqrt {x^{2}-1}}{x}}\right)}$	${\displaystyle \,\operatorname {arccsc} \left({\frac {1}{x}}\right)}$
arccos (x)	${\displaystyle \,\arcsin \left({\sqrt {1-x^{2}}}\right)}$	${\displaystyle \,\arccos(x)}$	${\displaystyle \,\arctan \left({\frac {1}{\sqrt {1+x^{2}}}}\right)}$	${\displaystyle \,\operatorname {arccot} \left({\frac {x}{\sqrt {x^{2}+1}}}\right)}$	${\displaystyle \,\operatorname {arcsec} \left({\frac {1}{x}}\right)}$	${\displaystyle \,\operatorname {arccsc} \left({\frac {\sqrt {x^{2}-1}}{x}}\right)}$
arctan (x)	${\displaystyle \,\arcsin \left(\pm {\frac {x}{\sqrt {1-x^{2}}}}\right)}$	${\displaystyle \,\arccos \left(\pm {\frac {\sqrt {1-x^{2}}}{x}}\right)}$	${\displaystyle \,\arctan(x)}$	${\displaystyle \,\operatorname {arccot} \left({\frac {1}{x}}\right)}$	${\displaystyle \,\operatorname {arcsec} \left(\pm {\sqrt {x^{2}-1}}\right)}$	${\displaystyle \,\operatorname {arccsc} \left(\pm {\frac {1}{\sqrt {x^{2}-1}}}\right)}$
arccot ​​(x)	${\displaystyle \,\arcsin \left({\frac {\sqrt {1-x^{2}}}{x}}\right)}$	${\displaystyle \,\arccos \left({\frac {x}{\sqrt {1-x^{2}}}}\right)}$	${\displaystyle \,\arctan \left({\frac {1}{x}}\right)}$	${\displaystyle \,\operatorname {arccot}(x)}$	${\displaystyle \,\operatorname {arcsec} \left({\frac {1}{\sqrt {x^{2}-1}}}\right)}$	${\displaystyle \,\operatorname {arccsc} \left({\sqrt {x^{2}-1}}\right)}$
arcsec (x)	${\displaystyle \,\arcsin \left({\frac {1}{\sqrt {1-x^{2}}}}\right)}$	${\displaystyle \,\arccos \left({\frac {1}{x}}\right)}$	${\displaystyle \,\arctan \left({\sqrt {1+x^{2}}}\right)}$	${\displaystyle \,\operatorname {arccot} \left({\frac {\sqrt {x^{2}+1}}{x}}\right)}$	${\displaystyle \,\operatorname {arcsec}(x)}$	${\displaystyle \,\operatorname {arccsc} \left({\frac {x}{\sqrt {x^{2}-1}}}\right)}$
arccsc (x)	${\displaystyle \,\arcsin \left({\frac {1}{x}}\right)}$	${\displaystyle \,\arccos \left(\pm {\frac {1}{\sqrt {1-x^{2}}}}\right)}$	${\displaystyle \,\arctan \left(\pm {\frac {\sqrt {1+x^{2}}}{x}}\right)}$	${\displaystyle \,\operatorname {arccot} \left(\pm {\sqrt {x^{2}+1}}\right)}$	${\displaystyle \,\operatorname {arcsec} \left(\pm {\frac {x}{\sqrt {x^{2}-1}}}\right)}$	${\displaystyle \,\operatorname {arccsc}(x)}$

If that is used, note that ${\displaystyle \,\pm }$

• ${\displaystyle \,\arcsin \,(x)\geq \ 0}$ For ${\displaystyle \,0\leq x\leq 1}$
• ${\displaystyle \,\arcsin \,(x)\leq \ 0}$ For ${\displaystyle \,-1\leq x\leq 0}$

• ${\displaystyle \,\arctan(x)\geq \ 0}$ For ${\displaystyle \,x\geq 0}$
• ${\displaystyle \,\arctan(x)\leq \ 0}$ For ${\displaystyle \,x\leq 0}$

• ${\displaystyle \,\operatorname {arccsc} \,(x)>\ 0}$ For ${\displaystyle \,x\geq 1}$
• ${\displaystyle \,\operatorname {arccsc} \,(x)<\ 0}$ For ${\displaystyle \,x\leq -1}$

When calculating from , and the calculated values ​​for from must be subtracted. ${\displaystyle \arccos(x)}$ ${\displaystyle \operatorname {arccot}(x)}$ ${\displaystyle \operatorname {arcsec}(x)}$ ${\displaystyle x<0}$ ${\displaystyle \pi }$

See also

• Trigonometric function
• Hyperbolic function
• Trigonometry formula collection

Web links

• Information on math online
• Eric W. Weisstein : Inverse Trigonometric Functions . In: MathWorld (English).