In mathematics, the inverse trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the trigonometric functions, under suitably restricted domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
Notation
There are several notations for the inverse trigonometric functions. The most common uses the prefix arc-: arcsin(x), arccos(x), arctan(x), etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Thus in the unit circle, the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is x" is the same as "the angle whose cosine is x", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin, acos, atan. Another common notation in English sources uses a superscript f−1: sin−1(x), cos−1(x), tan−1(x), etc., as introduced by John Herschel in 1813. This conflicts with the common syntax sin2(x), which denote exponentiation rather than function composition, and therefore may result in confusion between notation for the reciprocal (multiplicative inverse) and inverse function. The inverse function convention using a superscript in brackets is unambiguous but not commonly used: sin[−1](x), cos[−1](x), tan[−1](x). The ambiguity is mitigated by the use of special names for reciprocal trigonometric functions — for example, (cos(x))−1 = sec(x). Nevertheless, certain authors advise against using it, since it is ambiguous. Another problematic convention used by a small number of authors is an uppercase first letter, along with a “−1” superscript: Sin−1(x), Cos−1(x), Tan−1(x), etc. Although it is intended to avoid confusion with the reciprocal, which should be represented by sin−1(x), cos−1(x), etc., or, better, by sin−1 x, cos−1 x, etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA) use capitalised names for the standard trig functions. Since 2019, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.
Basic concepts
Principal values Since none of the six trigonometric functions are one-to-one, they must be restricted in order to have inverse functions. Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions. For example, using function in the sense of multivalued functions, just as the square root function y = x {\displaystyle y={\sqrt {x}}} could be defined from y 2 = x , {\displaystyle y^{2}=x,} the function y = arcsin ( x ) {\displaystyle y=\arcsin(x)} is defined so that sin ( y ) = x . {\displaystyle \sin(y)=x.} For a given real number x , {\displaystyle x,} with − 1 ≤ x ≤ 1 , {\displaystyle -1\leq x\leq 1,} there are multiple (in fact, countably infinitely many) numbers y {\displaystyle y} such that sin ( y ) = x {\displaystyle \sin(y)=x} ; for example, sin ( 0 ) = 0 , {\displaystyle \sin(0)=0,} but also sin ( π ) = 0 , {\displaystyle \sin(\pi )=0,} sin ( 2 π ) = 0 , {\displaystyle \sin(2\pi )=0,} etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x {\displaystyle x} in the domain, the expression arcsin ( x ) {\displaystyle \arcsin(x)} 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.
Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π ≤ y < 3 π 2 {\textstyle \pi \leq y<{\frac {3\pi }{2}}} ), because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, tan ( arcsec ( x ) ) = x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},} whereas with the range ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π 2 < y ≤ π {\textstyle {\frac {\pi }{2}}<y\leq \pi } ), we would have to write tan ( arcsec ( x ) ) = ± x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},} since tangent is nonnegative on 0 ≤ y < π 2 , {\textstyle 0\leq y<{\frac {\pi }{2}},} but nonpositive on π 2 < y ≤ π . {\textstyle {\frac {\pi }{2}}<y\leq \pi .} For a similar reason, the same authors define the range of arccosecant to be ( − π < y ≤ − π 2 {\textstyle (-\pi <y\leq -{\frac {\pi }{2}}} or 0 < y ≤ π 2 ) . {\textstyle 0<y\leq {\frac {\pi }{2}}).}
Domains If x is allowed to be a complex number, then the range of y applies only to its real part. The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians.
The symbol R = ( − ∞ , ∞ ) {\displaystyle \mathbb {R} =(-\infty ,\infty )} denotes the set of all real numbers and Z = { … , − 2 , − 1 , 0 , 1 , 2 , … } {\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}} denotes the set of all integers. The set of all integer multiples of π {\displaystyle \pi } is denoted by
π Z := { π n : n ∈ Z } = { … , − 2 π , − π , 0 , π , 2 π , … } . {\displaystyle \pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.}
The symbol ∖ {\displaystyle \,\setminus \,} denotes set subtraction so that, for instance, R ∖ ( − 1 , 1 ) = ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) {\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )} is the set of points in R {\displaystyle \mathbb {R} } (that is, real numbers) that are not in the interval ( − 1 , 1 ) . {\displaystyle (-1,1).}
The Minkowski sum notation π Z + ( 0 , π ) {\textstyle \pi \mathbb {Z} +(0,\pi )} and π Z + ( − π 2 , π 2 ) {\displaystyle \pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}} that is used above to concisely write the domains of cot , csc , tan , and sec {\displaystyle \cot ,\csc ,\tan ,{\text{ and }}\sec } is now explained. Domain of cotangent cot {\displaystyle \cot } and cosecant csc {\displaystyle \csc } : The domains of cot {\displaystyle \,\cot \,} and csc {\displaystyle \,\csc \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which sin θ ≠ 0 , {\displaystyle \sin \theta \neq 0,} i.e. all real numbers that are not of the form π n {\displaystyle \pi n} for some integer n , {\displaystyle n,}
π Z + ( 0 , π ) = ⋯ ∪ ( − 2 π , − π ) ∪ ( − π , 0 ) ∪ ( 0 , π ) ∪ ( π , 2 π ) ∪ ⋯ = R ∖ π Z {\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}}
Domain of tangent tan {\displaystyle \tan } and secant sec {\displaystyle \sec } : The domains of tan {\displaystyle \,\tan \,} and sec {\displaystyle \,\sec \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which cos θ ≠ 0 , {\displaystyle \cos \theta \neq 0,}
π Z + ( − π 2 , π 2 ) = ⋯ ∪ ( − 3 π 2 , − π 2 ) ∪ ( − π 2 , π 2 ) ∪ ( π 2 , 3 π 2 ) ∪ ⋯ = R ∖ ( π 2 + π Z ) {\displaystyle {\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}}
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 2 π : {\displaystyle 2\pi :}
Sine and cosecant begin their period at 2 π k − π 2 {\textstyle 2\pi k-{\frac {\pi }{2}}} (where k {\displaystyle k} is an integer), finish it at 2 π k + π 2 , {\textstyle 2\pi k+{\frac {\pi }{2}},} and then reverse themselves over 2 π k + π 2 {\textstyle 2\pi k+{\frac {\pi }{2}}} to 2 π k + 3 π 2 . {\textstyle 2\pi k+{\frac {3\pi }{2}}.}
Cosine and secant begin their period at 2 π k , {\displaystyle 2\pi k,} finish it at 2 π k + π . {\displaystyle 2\pi k+\pi .} and then reverse themselves over 2 π k + π {\displaystyle 2\pi k+\pi } to 2 π k + 2 π . {\displaystyle 2\pi k+2\pi .}
Tangent begins its period at 2 π k − π 2 , {\textstyle 2\pi k-{\frac {\pi }{2}},} finishes it at 2 π k + π 2 , {\textstyle 2\pi k+{\frac {\pi }{2}},} and then repeats it (forward) over 2 π k + π 2 {\textstyle 2\pi k+{\frac {\pi }{2}}} to 2 π k + 3 π 2 . {\textstyle 2\pi k+{\frac {3\pi }{2}}.}
Cotangent begins its period at 2 π k , {\displaystyle 2\pi k,} finishes it at 2 π k + π , {\displaystyle 2\pi k+\pi ,} and then repeats it (forward) over 2 π k + π {\displaystyle 2\pi k+\pi } to 2 π k + 2 π . {\displaystyle 2\pi k+2\pi .}
This periodicity is reflected in the general inverses, where k {\displaystyle k} 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 θ , {\displaystyle \theta ,} r , {\displaystyle r,} s , {\displaystyle s,} x , {\displaystyle x,} and y {\displaystyle y} all lie within appropriate ranges so that the relevant expressions below are well-defined. Note that "for some k ∈ Z {\displaystyle k\in \mathbb {Z} } " is just another way of saying "for some integer k . {\displaystyle k.} " The symbol ⟺ {\displaystyle \,\iff \,} 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 for more details and an example illustrating this concept).
where the first four solutions can be written in expanded form as:
For example, if cos θ = − 1 {\displaystyle \cos \theta =-1} then θ = π + 2 π k = − π + 2 π ( 1 + k ) {\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)} for some k ∈ Z . {\displaystyle k\in \mathbb {Z} .} While if sin θ = ± 1 {\displaystyle \sin \theta =\pm 1} then θ = π 2 + π k = − π 2 + π ( k + 1 ) {\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)} for some k ∈ Z , {\displaystyle k\in \mathbb {Z} ,} where k {\displaystyle k} will be even if sin θ = 1 {\displaystyle \sin \theta =1} and it will be odd if sin θ = − 1. {\displaystyle \sin \theta =-1.} The equations sec θ = − 1 {\displaystyle \sec \theta =-1} and csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} have the same solutions as cos θ = − 1 {\displaystyle \cos \theta =-1} and sin θ = ± 1 , {\displaystyle \sin \theta =\pm 1,} respectively. In all equations above except for those just solved (i.e. except for sin {\displaystyle \sin } / csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} and cos {\displaystyle \cos } / sec θ = − 1 {\displaystyle \sec \theta =-1} ), the integer k {\displaystyle k} in the solution's formula is uniquely determined by θ {\displaystyle \theta } (for fixed r , s , x , {\displaystyle r,s,x,} and y {\displaystyle y} ). With the help of integer parity
Parity ( h ) = { 0 if h is even 1 if h is odd {\displaystyle \operatorname {Parity} (h)={\begin{cases}0&{\text{if }}h{\text{ is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}}
it is possible to write a solution to cos θ = x {\displaystyle \cos \theta =x} that doesn't involve the "plus or minus" ± {\displaystyle \,\pm \,} symbol:
c o s θ = x {\displaystyle cos\;\theta =x\quad } if and only if θ = ( − 1 ) h arccos ( x ) + π h + π Parity ( h ) {\displaystyle \quad \theta =(-1)^{h}\arccos(x)+\pi h+\pi \operatorname {Parity} (h)\quad } for some h ∈ Z . {\displaystyle h\in \mathbb {Z} .}
And similarly for the secant function,
s e c θ = r {\displaystyle sec\;\theta =r\quad } if and only if θ = ( − 1 ) h arcsec ( r ) + π h + π Parity ( h ) {\displaystyle \quad \theta =(-1)^{h}\operatorname {arcsec}(r)+\pi h+\pi \operatorname {Parity} (h)\quad } for some h ∈ Z , {\displaystyle h\in \mathbb {Z} ,}
where π h + π Parity ( h ) {\displaystyle \pi h+\pi \operatorname {Parity} (h)} equals π h {\displaystyle \pi h} when the integer h {\displaystyle h} is even, and equals π h + π {\displaystyle \pi h+\pi } when it's odd.
Detailed example and explanation of the "plus or minus" symbol ± The solutions to cos θ = x {\displaystyle \cos \theta =x} and sec θ = x {\displaystyle \sec \theta =x} involve the "plus or minus" symbol ± , {\displaystyle \,\pm ,\,} whose meaning is now clarified. Only the solution to cos θ = x {\displaystyle \cos \theta =x} will be discussed since the discussion for sec θ = x {\displaystyle \sec \theta =x} is the same. We are given x {\displaystyle x} between − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} and we know that there is an angle θ {\displaystyle \theta } in some interval that satisfies cos θ = x . {\displaystyle \cos \theta =x.} We want to find this θ . {\displaystyle \theta .} The table above indicates that the solution is
θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} }
which is a shorthand way of saying that (at least) one of the following statement is true:
θ = arccos x + 2 π k {\displaystyle \,\theta =\arccos x+2\pi k\,} for some integer k , {\displaystyle k,} or
θ = − arccos x + 2 π k {\displaystyle \,\theta =-\arccos x+2\pi k\,} for some integer k . {\displaystyle k.}
As mentioned above, if arccos x = π {\displaystyle \,\arccos x=\pi \,} (which by definition only happens when x = cos π = − 1 {\displaystyle x=\cos \pi =-1} ) then both statements (1) and (2) hold, although with different values for the integer k {\displaystyle k} : if K {\displaystyle K} is the integer from statement (1), meaning that θ = π + 2 π K {\displaystyle \theta =\pi +2\pi K} holds, then the integer k {\displaystyle k} for statement (2) is K + 1 {\displaystyle K+1} (because θ = − π + 2 π ( 1 + K ) {\displaystyle \theta =-\pi +2\pi (1+K)} ). However, if x ≠ − 1 {\displaystyle x\neq -1} then the integer k {\displaystyle k} is unique and completely determined by θ . {\displaystyle \theta .} If arccos x = 0 {\displaystyle \,\arccos x=0\,} (which by definition only happens when x = cos 0 = 1 {\displaystyle x=\cos 0=1} ) then ± arccos x = 0 {\displaystyle \,\pm \arccos x=0\,} (because + arccos x = + 0 = 0 {\displaystyle \,+\arccos x=+0=0\,} and − arccos x = − 0 = 0 {\displaystyle \,-\arccos x=-0=0\,} so in both cases ± arccos x {\displaystyle \,\pm \arccos x\,} is equal to 0 {\displaystyle 0} ) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold). Having considered the cases arccos x = 0 {\displaystyle \,\arccos x=0\,} and arccos x = π , {\displaystyle \,\arccos x=\pi ,\,} we now focus on the case where arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and arccos x ≠ π , {\displaystyle \,\arccos x\neq \pi ,\,} So assume this from now on. The solution to cos θ = x {\displaystyle \cos \theta =x} is still
θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} }
which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and 0 < arccos x < π , {\displaystyle \,0<\arccos x<\pi ,\,} statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about θ {\displaystyle \theta } is needed to determine which one holds. For example, suppose that x = 0 {\displaystyle x=0} and that all that is known about θ {\displaystyle \theta } is that − π ≤ θ ≤ π {\displaystyle \,-\pi \leq \theta \leq \pi \,} (and nothing more is known). Then
arccos x = arccos 0 = π 2 {\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}} and moreover, in this particular case k = 0 {\displaystyle k=0} (for both the + {\displaystyle \,+\,} case and the − {\displaystyle \,-\,} case) and so consequently,
θ = ± arccos x + 2 π k = ± ( π 2 ) + 2 π ( 0 ) = ± π 2 . {\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.}
This means that θ {\displaystyle \theta } could be either π / 2 {\displaystyle \,\pi /2\,} or − π / 2. {\displaystyle \,-\pi /2.} Without additional information it is not possible to determine which of these values θ {\displaystyle \theta } has. An example of some additional information that could determine the value of θ {\displaystyle \theta } would be knowing that the angle is above the x {\displaystyle x} -axis (in which case θ = π / 2 {\displaystyle \theta =\pi /2} ) or alternatively, knowing that it is below the x {\displaystyle x} -axis (in which case θ = − π / 2 {\displaystyle \theta =-\pi /2} ).
Equal identical trigonometric functions The table below shows how two angles θ {\displaystyle \theta } and φ {\displaystyle \varphi } must be related if their values under a given trigonometric function are equal or negatives of each other.
The vertical double arrow ⇕ {\displaystyle \Updownarrow } in the last row indicates that θ {\displaystyle \theta } and φ {\displaystyle \varphi } satisfy | sin θ | = | sin φ