For a circle of radius 1, arcsin and arccos are the lengths of actual arcs determined by the quantities in question. Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin(x ) , arccos(x ) , arctan(x ) , etc.[6] (This convention is used throughout this article.) This notation arises from the following geometric relationships:[citation needed ] 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 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.[11] In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin , acos , atan .[12]
The notations sin−1 (x ) , cos−1 (x ) , tan−1 (x ) , etc., as introduced by John Herschel in 1813,[13] [14] are often used as well in English-language sources,[6] much more than the also established sin[−1] (x ) , cos[−1] (x ) , tan[−1] (x ) – conventions consistent with the notation of an inverse function , that is useful (for example) to define the multivalued version of each inverse trigonometric function: tan − 1 ( x ) = { arctan ( x ) + π k ∣ k ∈ Z } . {\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.} However, this might appear to conflict logically with the common semantics for expressions such as sin2 (x ) (although only sin2 x , without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal (multiplicative inverse ) and inverse function .[15]
The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, (cos(x ))−1 = sec(x ) . Nevertheless, certain authors advise against using it, since it is ambiguous.[6] [16] Another precarious convention used by a small number of authors is to use an uppercase first letter, along with a “−1 ” superscript: Sin−1 (x ) , Cos−1 (x ) , Tan−1 (x ) , etc.[17] 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 those very same capitalised representations for the standard trig functions, whereas others (Python , SymPy , NumPy , Matlab , MAPLE , etc.) use lower-case.
Hence, since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.
The points labelled 1 , Sec(θ ) , Csc(θ ) represent the length of the line segment from the origin to that point. Sin(θ ) , Tan(θ ) , and 1 are the heights to the line starting from the x -axis, while Cos(θ ) , 1 , and Cot(θ ) are lengths along the x -axis starting from the origin. Principal values edit 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.
Name Usual notation Definition Domain of x {\displaystyle x} for real result Range of usual principal value (radians ) Range of usual principal value (degrees ) arcsine y = arcsin ( x ) {\displaystyle y=\arcsin(x)} x = sin (y ) − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} − π 2 ≤ y ≤ π 2 {\displaystyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}} − 90 ∘ ≤ y ≤ 90 ∘ {\displaystyle -90^{\circ }\leq y\leq 90^{\circ }} arccosine y = arccos ( x ) {\displaystyle y=\arccos(x)} x = cos (y ) − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} 0 ≤ y ≤ π {\displaystyle 0\leq y\leq \pi } 0 ∘ ≤ y ≤ 180 ∘ {\displaystyle 0^{\circ }\leq y\leq 180^{\circ }} arctangent y = arctan ( x ) {\displaystyle y=\arctan(x)} x = tan (y ) all real numbers − π 2 < y < π 2 {\displaystyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}} − 90 ∘ < y < 90 ∘ {\displaystyle -90^{\circ }<y<90^{\circ }} arccotangent y = arccot ( x ) {\displaystyle y=\operatorname {arccot}(x)} x = cot (y ) all real numbers 0 < y < π {\displaystyle 0<y<\pi } 0 ∘ < y < 180 ∘ {\displaystyle 0^{\circ }<y<180^{\circ }} arcsecant y = arcsec ( x ) {\displaystyle y=\operatorname {arcsec}(x)} x = sec (y ) | x | ≥ 1 {\displaystyle {\left\vert x\right\vert }\geq 1} 0 ≤ y < π 2 or π 2 < y ≤ π {\displaystyle 0\leq y<{\frac {\pi }{2}}{\text{ or }}{\frac {\pi }{2}}<y\leq \pi } 0 ∘ ≤ y < 90 ∘ or 90 ∘ < y ≤ 180 ∘ {\displaystyle 0^{\circ }\leq y<90^{\circ }{\text{ or }}90^{\circ }<y\leq 180^{\circ }} arccosecant y = arccsc ( x ) {\displaystyle y=\operatorname {arccsc}(x)} x = csc (y ) | x | ≥ 1 {\displaystyle {\left\vert x\right\vert }\geq 1} − π 2 ≤ y < 0 or 0 < y ≤ π 2 {\displaystyle -{\frac {\pi }{2}}\leq y<0{\text{ or }}0<y\leq {\frac {\pi }{2}}} − 90 ∘ ≤ y < 0 ∘ or 0 ∘ < y ≤ 90 ∘ {\displaystyle -90^{\circ }\leq y<0^{\circ }{\text{ or }}0^{\circ }<y\leq 90^{\circ }}
Note: Some authors[citation needed ] define the range of arcsecant to be ( 0 ≤ y < π 2 or π ≤ y < 3 π 2 ) {\textstyle (0\leq y<{\frac {\pi }{2}}{\text{ or }}\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 or π 2 < y ≤ π ) {\textstyle (0\leq y<{\frac {\pi }{2}}{\text{ or }}{\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 edit If x {\displaystyle x} is allowed to be a complex number , then the range of y {\displaystyle 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 .
Name Symbol Domain Image/Range Inverse function Domain Image of principal values sine sin {\displaystyle \sin } : {\displaystyle :} R {\displaystyle \mathbb {R} } → {\displaystyle \to } [ − 1 , 1 ] {\displaystyle [-1,1]} arcsin {\displaystyle \arcsin } : {\displaystyle :} [ − 1 , 1 ] {\displaystyle [-1,1]} → {\displaystyle \to } [ − π 2 , π 2 ] {\displaystyle \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]} cosine cos {\displaystyle \cos } : {\displaystyle :} R {\displaystyle \mathbb {R} } → {\displaystyle \to } [ − 1 , 1 ] {\displaystyle [-1,1]} arccos {\displaystyle \arccos } : {\displaystyle :} [ − 1 , 1 ] {\displaystyle [-1,1]} → {\displaystyle \to } [ 0 , π ] {\displaystyle [0,\pi ]} tangent tan {\displaystyle \tan } : {\displaystyle :} π Z + ( − π 2 , π 2 ) {\displaystyle \pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)} → {\displaystyle \to } R {\displaystyle \mathbb {R} } arctan {\displaystyle \arctan } : {\displaystyle :} R {\displaystyle \mathbb {R} } → {\displaystyle \to } ( − π 2 , π 2 ) {\displaystyle \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)} cotangent cot {\displaystyle \cot } : {\displaystyle :} π Z + ( 0 , π ) {\displaystyle \pi \mathbb {Z} +(0,\pi )} → {\displaystyle \to } R {\displaystyle \mathbb {R} } arccot {\displaystyle \operatorname {arccot} } : {\displaystyle :} R {\displaystyle \mathbb {R} } → {\displaystyle \to } ( 0 , π ) {\displaystyle (0,\pi )} secant sec {\displaystyle \sec } : {\displaystyle :} π Z + ( − π 2 , π 2 ) {\displaystyle \pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)} → {\displaystyle \to } R ∖ ( − 1 , 1 ) {\displaystyle \mathbb {R} \setminus (-1,1)} arcsec {\displaystyle \operatorname {arcsec} } : {\displaystyle :} R ∖ ( − 1 , 1 ) {\displaystyle \mathbb {R} \setminus (-1,1)} → {\displaystyle \to } [ 0 , π ] ∖ { π 2 } {\displaystyle [\,0,\;\pi \,]\;\;\;\setminus \left\{{\tfrac {\pi }{2}}\right\}} cosecant csc {\displaystyle \csc } : {\displaystyle :} π Z + ( 0 , π ) {\displaystyle \pi \mathbb {Z} +(0,\pi )} → {\displaystyle \to } R ∖ ( − 1 , 1 ) {\displaystyle \mathbb {R} \setminus (-1,1)} arccsc {\displaystyle \operatorname {arccsc} } : {\displaystyle :} R ∖ ( − 1 , 1 ) {\displaystyle \mathbb {R} \setminus (-1,1)} → {\displaystyle \to } [ − π 2 , π 2 ] ∖ { 0 } {\displaystyle \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}}
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 edit 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 . The expression "LHS ⟺ {\displaystyle \,\iff \,} RHS" indicates that either (a) the left hand side (i.e. LHS) and right hand side (i.e. RHS) are both true, or else (b) the left hand side and right hand side are both false; there is no option (c) (e.g. it is not possible for the LHS statement to be true and also simultaneously for the RHS statement to be false), because otherwise "LHS ⟺ {\displaystyle \,\iff \,} RHS" would not have been written (see this footnote[note 1] for an example illustrating this concept).
Equation if and only if Solution sin θ = y {\displaystyle \sin \theta =y} ⟺ {\displaystyle \iff } θ = {\displaystyle \theta =\,} ( − 1 ) k {\displaystyle (-1)^{k}} arcsin ( y ) {\displaystyle \arcsin(y)} + {\displaystyle +} π k {\displaystyle \pi k} for some k ∈ Z {\displaystyle k\in \mathbb {Z} } csc θ = r {\displaystyle \csc \theta =r} ⟺ {\displaystyle \iff } θ = {\displaystyle \theta =\,} ( − 1 ) k {\displaystyle (-1)^{k}} arccsc ( r ) {\displaystyle \operatorname {arccsc}(r)} + {\displaystyle +} π k {\displaystyle \pi k} for some k ∈ Z {\displaystyle k\in \mathbb {Z} } cos θ = x {\displaystyle \cos \theta =x} ⟺ {\displaystyle \iff } θ = {\displaystyle \theta =\,} ± {\displaystyle \pm \,} arccos ( x ) {\displaystyle \arccos(x)} + {\displaystyle +} 2 {\displaystyle 2} π k {\displaystyle \pi k} for some k ∈ Z {\displaystyle k\in \mathbb {Z} } sec θ = r {\displaystyle \sec \theta =r} ⟺ {\displaystyle \iff } θ = {\displaystyle \theta =\,} ± {\displaystyle \pm \,} arcsec ( r ) {\displaystyle \operatorname {arcsec}(r)} + {\displaystyle +} 2 {\displaystyle 2} π k {\displaystyle \pi k} for some k ∈ Z {\displaystyle k\in \mathbb {Z} } tan θ = s {\displaystyle \tan \theta =s} ⟺ {\displaystyle \iff } θ = {\displaystyle \theta =\,} arctan ( s ) {\displaystyle \arctan(s)} + {\displaystyle +} π k {\displaystyle \pi k} for some k ∈ Z {\displaystyle k\in \mathbb {Z} } cot θ = r {\displaystyle \cot \theta =r} ⟺ {\displaystyle \iff } θ = {\displaystyle \theta =\,} arccot ( r ) {\displaystyle \operatorname {arccot}(r)} + {\displaystyle +} π k {\displaystyle \pi k} for some k ∈ Z {\displaystyle k\in \mathbb {Z} }
where the first four solutions can be written in expanded form as:
Equation if and only if Solution sin θ = y {\displaystyle \sin \theta =y} ⟺ {\displaystyle \iff } θ = arcsin ( y ) + 2 π h {\displaystyle \theta =\;\;\;\,\arcsin(y)+2\pi h} or θ = − arcsin ( y ) + 2 π h + π {\displaystyle \theta =-\arcsin(y)+2\pi h+\pi } for some h ∈ Z {\displaystyle h\in \mathbb {Z} } csc θ = r {\displaystyle \csc \theta =r} ⟺ {\displaystyle \iff } θ = arccsc ( y ) + 2 π h {\displaystyle \theta =\;\;\;\,\operatorname {arccsc}(y)+2\pi h} or θ = − arccsc ( y ) + 2 π h + π {\displaystyle \theta =-\operatorname {arccsc}(y)+2\pi h+\pi } for some h ∈ Z {\displaystyle h\in \mathbb {Z} } cos θ = x {\displaystyle \cos \theta =x} ⟺ {\displaystyle \iff } θ = arccos ( y ) + 2 π k {\displaystyle \theta =\;\;\;\,\arccos(y)+2\pi k} or θ = − arccos ( y ) + 2 π k {\displaystyle \theta =-\arccos(y)+2\pi k} for some k ∈ Z {\displaystyle k\in \mathbb {Z} } sec θ = r {\displaystyle \sec \theta =r} ⟺ {\displaystyle \iff } θ = arcsec ( y ) + 2 π k {\displaystyle \theta =\;\;\;\,\operatorname {arcsec}(y)+2\pi k} or θ = − arcsec ( y ) + 2 π k {\displaystyle \theta =-\operatorname {arcsec}(y)+2\pi k} for some k ∈ Z {\displaystyle k\in \mathbb {Z} }
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} ).
Detailed example and explanation of the "plus or minus" symbol ± edit 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 edit 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.
Equation if and only if Solution (for some k ∈ Z {\displaystyle k\in \mathbb {Z} } ) Also a solution to − sin θ = sin φ {\displaystyle {\phantom {-}}\sin \theta =\sin \varphi } ⟺ {\displaystyle \iff } θ = ( − 1 ) k φ + 2 π k + π {\displaystyle \theta ={\phantom {\quad }}(-1)^{k}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}} − csc θ = csc φ {\displaystyle {\phantom {-}}\csc \theta =\csc \varphi } − cos θ = cos φ {\displaystyle {\phantom {-}}\cos \theta =\cos \varphi } ⟺ {\displaystyle \iff } θ = − 1 ± φ + 2 π k + π {\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +2\pi k{\phantom {+\pi }}} − sec θ = sec φ {\displaystyle {\phantom {-}}\sec \theta =\sec \varphi } − tan θ = tan φ {\displaystyle {\phantom {-}}\tan \theta =\tan \varphi } ⟺ {\displaystyle \iff } θ = ( − 1 ) k + 1 φ + 2 π k + π {\displaystyle \theta ={\phantom {(-1)^{k+1}}}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}} − cot θ = cot φ {\displaystyle {\phantom {-}}\cot \theta =\cot \varphi } − sin θ = sin φ {\displaystyle -\sin \theta =\sin \varphi } ⟺ {\displaystyle \iff } θ = ( − 1 ) k + 1 φ + 2 π k + π {\displaystyle \theta =(-1)^{k+1}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}} − csc θ = csc φ {\displaystyle -\csc \theta =\csc \varphi } − cos θ = cos φ {\displaystyle -\cos \theta =\cos \varphi } ⟺ {\displaystyle \iff } θ = − 1 ± φ + 2 π k + π + π {\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +2\pi k+\pi {\phantom {+\pi }}} − sec θ = sec φ {\displaystyle -\sec \theta =\sec \varphi } − tan θ = tan φ {\displaystyle -\tan \theta =\tan \varphi } ⟺ {\displaystyle \iff } θ = − 1 − φ + 2 π k + π {\displaystyle \theta ={\phantom {-1\quad }}-\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}} − cot θ = cot φ {\displaystyle -\cot \theta =\cot \varphi } − | sin θ | = | sin φ | ⇕ − | cos θ | = | cos φ | {\displaystyle {\begin{aligned}{\phantom {-}}\left|\sin \theta \right|&=\left|\sin \varphi \right|\\&\Updownarrow \\{\phantom {-}}\left|\cos \theta \right|&=\left|\cos \varphi \right|\end{aligned}}} ⟺ {\displaystyle \iff } θ = − 1 ± φ + 2 π k + π {\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +{\phantom {2}}\pi k{\phantom {+\pi }}} − | tan θ | = | tan φ | | csc θ | = | csc φ | | sec θ | = | sec φ | | cot θ | = | cot φ | {\displaystyle {\begin{aligned}{\phantom {-}}\left|\tan \theta \right|&=\left|\tan \varphi \right|\\\left|\csc \theta \right|&=\left|\csc \varphi \right|\\\left|\sec \theta \right|&=\left|\sec \varphi \right|\\\left|\cot \theta \right|&=\left|\cot \varphi \right|\end{aligned}}}
The vertical double arrow ⇕ {\displaystyle \Updownarrow } in the last row indicates that θ {\displaystyle \theta } and φ {\displaystyle \varphi } satisfy | sin θ | = | sin φ | {\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|} if and only if they satisfy | cos θ | = | cos φ | . {\displaystyle \left|\cos \theta \right|=\left|\cos \varphi \right|.}
Set of all solutions to elementary trigonometric equations Thus given a single solution θ {\displaystyle \theta } to an elementary trigonometric equation ( sin θ = y {\displaystyle \sin \theta =y} is such an equation, for instance, and because sin ( arcsin y ) = y {\displaystyle \sin(\arcsin y)=y} always holds, θ := arcsin y {\displaystyle \theta :=\arcsin y} is always a solution), the set of all solutions to it are:
If θ {\displaystyle \theta } solves then Set of all solutions (in terms of θ {\displaystyle \theta } ) sin θ = y {\displaystyle \;\sin \theta =y} then { φ : sin φ = y } = {\displaystyle \{\varphi :\sin \varphi =y\}=\,} ( θ {\displaystyle (\theta } + 2 {\displaystyle \,+\,2} π Z ) {\displaystyle \pi \mathbb {Z} )} ∪ ( − θ {\displaystyle \,\cup \,(-\theta } − π {\displaystyle -\pi } + 2 π Z ) {\displaystyle +2\pi \mathbb {Z} )} csc θ = r {\displaystyle \;\csc \theta =r} then { φ : csc φ = r } = {\displaystyle \{\varphi :\csc \varphi =r\}=\,} ( θ {\displaystyle (\theta } + 2 {\displaystyle \,+\,2} π Z ) {\displaystyle \pi \mathbb {Z} )} ∪ ( − θ {\displaystyle \,\cup \,(-\theta } − π {\displaystyle -\pi } + 2 π Z ) {\displaystyle +2\pi \mathbb {Z} )} cos θ = x {\displaystyle \;\cos \theta =x} then { φ : cos φ = x } = {\displaystyle \{\varphi :\cos \varphi =x\}=\,} ( θ {\displaystyle (\theta } + 2 {\displaystyle \,+\,2} π Z ) {\displaystyle \pi \mathbb {Z} )} ∪ ( − θ {\displaystyle \,\cup \,(-\theta } + 2 π Z ) {\displaystyle +2\pi \mathbb {Z} )} sec θ = r {\displaystyle \;\sec \theta =r} then { φ : sec φ = r } = {\displaystyle \{\varphi :\sec \varphi =r\}=\,} ( θ {\displaystyle (\theta } + 2 {\displaystyle \,+\,2} π Z ) {\displaystyle \pi \mathbb {Z} )} ∪ ( − θ {\displaystyle \,\cup \,(-\theta } + 2 π Z ) {\displaystyle +2\pi \mathbb {Z} )} tan θ = s {\displaystyle \;\tan \theta =s} then { φ : tan φ = s } = {\displaystyle \{\varphi :\tan \varphi =s\}=\,} θ {\displaystyle \theta } + {\displaystyle \,+\,} π Z {\displaystyle \pi \mathbb {Z} } cot θ = r {\displaystyle \;\cot \theta =r} then { φ : cot φ = r } = {\displaystyle \{\varphi :\cot \varphi =r\}=\,} θ {\displaystyle \theta } + {\displaystyle \,+\,} π Z {\displaystyle \pi \mathbb {Z} }
inverse, trigonometric, functions, arctangent, redirects, here, music, event, arctangent, festival, mathematics, inverse, trigonometric, functions, occasionally, also, called, arcus, functions, antitrigonometric, functions, cyclometric, functions, inverse, fun. Arctangent redirects here For the music event see ArcTanGent Festival In mathematics the inverse trigonometric functions occasionally also called arcus functions 1 2 3 4 5 antitrigonometric functions 6 or cyclometric functions 7 8 9 are the inverse functions of the trigonometric functions with suitably restricted domains Specifically they are the inverses of the sine cosine tangent cotangent secant and cosecant functions 10 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 Contents 1 Notation 2 Basic concepts 2 1 Principal values 2 1 1 Domains 2 2 Solutions to elementary trigonometric equations 2 2 1 Detailed example and explanation of the plus or minus symbol 2 2 2 Equal identical trigonometric functions 2 3 Transforming equations 2 4 Relationships between trigonometric functions and inverse trigonometric functions 2 5 Relationships among the inverse trigonometric functions 2 6 Arctangent addition formula 3 In calculus 3 1 Derivatives of inverse trigonometric functions 3 2 Expression as definite integrals 3 3 Infinite series 3 3 1 Continued fractions for arctangent 3 4 Indefinite integrals of inverse trigonometric functions 3 4 1 Example 4 Extension to the complex plane 4 1 Logarithmic forms 4 1 1 Generalization 4 1 2 Example proof 5 Applications 5 1 Finding the angle of a right triangle 5 2 In computer science and engineering 5 2 1 Two argument variant of arctangent 5 2 2 Arctangent function with location parameter 5 2 3 Numerical accuracy 6 See also 7 Notes 8 References 9 External linksNotation edit nbsp For a circle of radius 1 arcsin and arccos are the lengths of actual arcs determined by the quantities in question See also Trigonometric functions Notation Several notations for the inverse trigonometric functions exist The most common convention is to name inverse trigonometric functions using an arc prefix arcsin x arccos x arctan x etc 6 This convention is used throughout this article This notation arises from the following geometric relationships citation needed when measuring in radians an angle of 8 radians will correspond to an arc whose length is r8 where r is the radius of the circle Thus in the unit circle 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 11 In computer programming languages the inverse trigonometric functions are often called by the abbreviated forms asin acos atan 12 The notations sin 1 x cos 1 x tan 1 x etc as introduced by John Herschel in 1813 13 14 are often used as well in English language sources 6 much more than the also established sin 1 x cos 1 x tan 1 x conventions consistent with the notation of an inverse function that is useful for example to define the multivalued version of each inverse trigonometric function tan 1 x arctan x p k k Z displaystyle tan 1 x arctan x pi k mid k in mathbb Z nbsp However this might appear to conflict logically with the common semantics for expressions such as sin2 x although only sin2 x without parentheses is the really common use which refer to numeric power rather than function composition and therefore may result in confusion between notation for the reciprocal multiplicative inverse and inverse function 15 The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name for example cos x 1 sec x Nevertheless certain authors advise against using it since it is ambiguous 6 16 Another precarious convention used by a small number of authors is to use an uppercase first letter along with a 1 superscript Sin 1 x Cos 1 x Tan 1 x etc 17 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 those very same capitalised representations for the standard trig functions whereas others Python SymPy NumPy Matlab MAPLE etc use lower case Hence since 2009 the ISO 80000 2 standard has specified solely the arc prefix for the inverse functions Basic concepts edit nbsp The points labelled 1 Sec 8 Csc 8 represent the length of the line segment from the origin to that point Sin 8 Tan 8 and 1 are the heights to the line starting from the x axis while Cos 8 1 and Cot 8 are lengths along the x axis starting from the origin Principal values edit 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 nbsp could be defined from y 2 x displaystyle y 2 x nbsp the function y arcsin x displaystyle y arcsin x nbsp is defined so that sin y x displaystyle sin y x nbsp For a given real number x displaystyle x nbsp with 1 x 1 displaystyle 1 leq x leq 1 nbsp there are multiple in fact countably infinitely many numbers y displaystyle y nbsp such that sin y x displaystyle sin y x nbsp for example sin 0 0 displaystyle sin 0 0 nbsp but also sin p 0 displaystyle sin pi 0 nbsp sin 2 p 0 displaystyle sin 2 pi 0 nbsp etc When only one value is desired the function may be restricted to its principal branch With this restriction for each x displaystyle x nbsp in the domain the expression arcsin x displaystyle arcsin x nbsp 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 Name Usual notation Definition Domain of x displaystyle x nbsp for real result Range of usual principal value radians Range of usual principal value degrees arcsine y arcsin x displaystyle y arcsin x nbsp x sin y 1 x 1 displaystyle 1 leq x leq 1 nbsp p 2 y p 2 displaystyle frac pi 2 leq y leq frac pi 2 nbsp 90 y 90 displaystyle 90 circ leq y leq 90 circ nbsp arccosine y arccos x displaystyle y arccos x nbsp x cos y 1 x 1 displaystyle 1 leq x leq 1 nbsp 0 y p displaystyle 0 leq y leq pi nbsp 0 y 180 displaystyle 0 circ leq y leq 180 circ nbsp arctangent y arctan x displaystyle y arctan x nbsp x tan y all real numbers p 2 lt y lt p 2 displaystyle frac pi 2 lt y lt frac pi 2 nbsp 90 lt y lt 90 displaystyle 90 circ lt y lt 90 circ nbsp arccotangent y arccot x displaystyle y operatorname arccot x nbsp x cot y all real numbers 0 lt y lt p displaystyle 0 lt y lt pi nbsp 0 lt y lt 180 displaystyle 0 circ lt y lt 180 circ nbsp arcsecant y arcsec x displaystyle y operatorname arcsec x nbsp x sec y x 1 displaystyle left vert x right vert geq 1 nbsp 0 y lt p 2 or p 2 lt y p displaystyle 0 leq y lt frac pi 2 text or frac pi 2 lt y leq pi nbsp 0 y lt 90 or 90 lt y 180 displaystyle 0 circ leq y lt 90 circ text or 90 circ lt y leq 180 circ nbsp arccosecant y arccsc x displaystyle y operatorname arccsc x nbsp x csc y x 1 displaystyle left vert x right vert geq 1 nbsp p 2 y lt 0 or 0 lt y p 2 displaystyle frac pi 2 leq y lt 0 text or 0 lt y leq frac pi 2 nbsp 90 y lt 0 or 0 lt y 90 displaystyle 90 circ leq y lt 0 circ text or 0 circ lt y leq 90 circ nbsp Note Some authors citation needed define the range of arcsecant to be 0 y lt p 2 or p y lt 3 p 2 textstyle 0 leq y lt frac pi 2 text or pi leq y lt frac 3 pi 2 nbsp 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 nbsp whereas with the range 0 y lt p 2 or p 2 lt y p textstyle 0 leq y lt frac pi 2 text or frac pi 2 lt y leq pi nbsp we would have to write tan arcsec x x 2 1 displaystyle tan operatorname arcsec x pm sqrt x 2 1 nbsp since tangent is nonnegative on 0 y lt p 2 textstyle 0 leq y lt frac pi 2 nbsp but nonpositive on p 2 lt y p textstyle frac pi 2 lt y leq pi nbsp For a similar reason the same authors define the range of arccosecant to be p lt y p 2 textstyle pi lt y leq frac pi 2 nbsp or 0 lt y p 2 textstyle 0 lt y leq frac pi 2 nbsp Domains edit If x displaystyle x nbsp is allowed to be a complex number then the range of y displaystyle y nbsp 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 Name Symbol Domain Image Range Inverse function Domain Image of principal valuessine sin displaystyle sin nbsp displaystyle nbsp R displaystyle mathbb R nbsp displaystyle to nbsp 1 1 displaystyle 1 1 nbsp arcsin displaystyle arcsin nbsp displaystyle nbsp 1 1 displaystyle 1 1 nbsp displaystyle to nbsp p 2 p 2 displaystyle left tfrac pi 2 tfrac pi 2 right nbsp cosine cos displaystyle cos nbsp displaystyle nbsp R displaystyle mathbb R nbsp displaystyle to nbsp 1 1 displaystyle 1 1 nbsp arccos displaystyle arccos nbsp displaystyle nbsp 1 1 displaystyle 1 1 nbsp displaystyle to nbsp 0 p displaystyle 0 pi nbsp tangent tan displaystyle tan nbsp displaystyle nbsp p Z p 2 p 2 displaystyle pi mathbb Z left tfrac pi 2 tfrac pi 2 right nbsp displaystyle to nbsp R displaystyle mathbb R nbsp arctan displaystyle arctan nbsp displaystyle nbsp R displaystyle mathbb R nbsp displaystyle to nbsp p 2 p 2 displaystyle left tfrac pi 2 tfrac pi 2 right nbsp cotangent cot displaystyle cot nbsp displaystyle nbsp p Z 0 p displaystyle pi mathbb Z 0 pi nbsp displaystyle to nbsp R displaystyle mathbb R nbsp arccot displaystyle operatorname arccot nbsp displaystyle nbsp R displaystyle mathbb R nbsp displaystyle to nbsp 0 p displaystyle 0 pi nbsp secant sec displaystyle sec nbsp displaystyle nbsp p Z p 2 p 2 displaystyle pi mathbb Z left tfrac pi 2 tfrac pi 2 right nbsp displaystyle to nbsp R 1 1 displaystyle mathbb R setminus 1 1 nbsp arcsec displaystyle operatorname arcsec nbsp displaystyle nbsp R 1 1 displaystyle mathbb R setminus 1 1 nbsp displaystyle to nbsp 0 p p 2 displaystyle 0 pi setminus left tfrac pi 2 right nbsp cosecant csc displaystyle csc nbsp displaystyle nbsp p Z 0 p displaystyle pi mathbb Z 0 pi nbsp displaystyle to nbsp R 1 1 displaystyle mathbb R setminus 1 1 nbsp arccsc displaystyle operatorname arccsc nbsp displaystyle nbsp R 1 1 displaystyle mathbb R setminus 1 1 nbsp displaystyle to nbsp p 2 p 2 0 displaystyle left tfrac pi 2 tfrac pi 2 right setminus 0 nbsp The symbol R displaystyle mathbb R infty infty nbsp denotes the set of all real numbers and Z 2 1 0 1 2 displaystyle mathbb Z ldots 2 1 0 1 2 ldots nbsp denotes the set of all integers The set of all integer multiples of p displaystyle pi nbsp is denoted byp Z p n n Z 2 p p 0 p 2 p displaystyle pi mathbb Z pi n n in mathbb Z ldots 2 pi pi 0 pi 2 pi ldots nbsp The symbol displaystyle setminus nbsp denotes set subtraction so that for instance R 1 1 1 1 displaystyle mathbb R setminus 1 1 infty 1 cup 1 infty nbsp is the set of points in R displaystyle mathbb R nbsp that is real numbers that are not in the interval 1 1 displaystyle 1 1 nbsp The Minkowski sum notation p Z 0 p textstyle pi mathbb Z 0 pi nbsp and p Z p 2 p 2 displaystyle pi mathbb Z bigl tfrac pi 2 tfrac pi 2 bigr nbsp that is used above to concisely write the domains of cot csc tan and sec displaystyle cot csc tan text and sec nbsp is now explained Domain of cotangent cot displaystyle cot nbsp and cosecant csc displaystyle csc nbsp The domains of cot displaystyle cot nbsp and csc displaystyle csc nbsp are the same They are the set of all angles 8 displaystyle theta nbsp at which sin 8 0 displaystyle sin theta neq 0 nbsp i e all real numbers that are not of the form p n displaystyle pi n nbsp for some integer n displaystyle n nbsp p Z 0 p 2 p p p 0 0 p p 2 p R p Z displaystyle begin aligned pi mathbb Z 0 pi amp cdots cup 2 pi pi cup pi 0 cup 0 pi cup pi 2 pi cup cdots amp mathbb R setminus pi mathbb Z end aligned nbsp Domain of tangent tan displaystyle tan nbsp and secant sec displaystyle sec nbsp The domains of tan displaystyle tan nbsp and sec displaystyle sec nbsp are the same They are the set of all angles 8 displaystyle theta nbsp at which cos 8 0 displaystyle cos theta neq 0 nbsp p Z p 2 p 2 3 p 2 p 2 p 2 p 2 p 2 3 p 2 R p 2 p Z displaystyle begin aligned pi mathbb Z left tfrac pi 2 tfrac pi 2 right amp 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 amp mathbb R setminus left tfrac pi 2 pi mathbb Z right end aligned nbsp Solutions to elementary trigonometric equations edit 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 p displaystyle 2 pi nbsp Sine and cosecant begin their period at 2 p k p 2 textstyle 2 pi k frac pi 2 nbsp where k displaystyle k nbsp is an integer finish it at 2 p k p 2 textstyle 2 pi k frac pi 2 nbsp and then reverse themselves over 2 p k p 2 textstyle 2 pi k frac pi 2 nbsp to 2 p k 3 p 2 textstyle 2 pi k frac 3 pi 2 nbsp Cosine and secant begin their period at 2 p k displaystyle 2 pi k nbsp finish it at 2 p k p displaystyle 2 pi k pi nbsp and then reverse themselves over 2 p k p displaystyle 2 pi k pi nbsp to 2 p k 2 p displaystyle 2 pi k 2 pi nbsp Tangent begins its period at 2 p k p 2 textstyle 2 pi k frac pi 2 nbsp finishes it at 2 p k p 2 textstyle 2 pi k frac pi 2 nbsp and then repeats it forward over 2 p k p 2 textstyle 2 pi k frac pi 2 nbsp to 2 p k 3 p 2 textstyle 2 pi k frac 3 pi 2 nbsp Cotangent begins its period at 2 p k displaystyle 2 pi k nbsp finishes it at 2 p k p displaystyle 2 pi k pi nbsp and then repeats it forward over 2 p k p displaystyle 2 pi k pi nbsp to 2 p k 2 p displaystyle 2 pi k 2 pi nbsp This periodicity is reflected in the general inverses where k displaystyle k nbsp 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 8 displaystyle theta nbsp r displaystyle r nbsp s displaystyle s nbsp x displaystyle x nbsp and y displaystyle y nbsp 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 nbsp is just another way of saying for some integer k displaystyle k nbsp The symbol displaystyle iff nbsp is logical equality The expression LHS displaystyle iff nbsp RHS indicates that either a the left hand side i e LHS and right hand side i e RHS are both true or else b the left hand side and right hand side are both false there is no option c e g it is not possible for the LHS statement to be true and also simultaneously for the RHS statement to be false because otherwise LHS displaystyle iff nbsp RHS would not have been written see this footnote note 1 for an example illustrating this concept Equation if and only if Solutionsin 8 y displaystyle sin theta y nbsp displaystyle iff nbsp 8 displaystyle theta nbsp 1 k displaystyle 1 k nbsp arcsin y displaystyle arcsin y nbsp displaystyle nbsp p k displaystyle pi k nbsp for some k Z displaystyle k in mathbb Z nbsp csc 8 r displaystyle csc theta r nbsp displaystyle iff nbsp 8 displaystyle theta nbsp 1 k displaystyle 1 k nbsp arccsc r displaystyle operatorname arccsc r nbsp displaystyle nbsp p k displaystyle pi k nbsp for some k Z displaystyle k in mathbb Z nbsp cos 8 x displaystyle cos theta x nbsp displaystyle iff nbsp 8 displaystyle theta nbsp displaystyle pm nbsp arccos x displaystyle arccos x nbsp displaystyle nbsp 2 displaystyle 2 nbsp p k displaystyle pi k nbsp for some k Z displaystyle k in mathbb Z nbsp sec 8 r displaystyle sec theta r nbsp displaystyle iff nbsp 8 displaystyle theta nbsp displaystyle pm nbsp arcsec r displaystyle operatorname arcsec r nbsp displaystyle nbsp 2 displaystyle 2 nbsp p k displaystyle pi k nbsp for some k Z displaystyle k in mathbb Z nbsp tan 8 s displaystyle tan theta s nbsp displaystyle iff nbsp 8 displaystyle theta nbsp arctan s displaystyle arctan s nbsp displaystyle nbsp p k displaystyle pi k nbsp for some k Z displaystyle k in mathbb Z nbsp cot 8 r displaystyle cot theta r nbsp displaystyle iff nbsp 8 displaystyle theta nbsp arccot r displaystyle operatorname arccot r nbsp displaystyle nbsp p k displaystyle pi k nbsp for some k Z displaystyle k in mathbb Z nbsp where the first four solutions can be written in expanded form as Equation if and only if Solutionsin 8 y displaystyle sin theta y nbsp displaystyle iff nbsp 8 arcsin y 2 p h displaystyle theta arcsin y 2 pi h nbsp or 8 arcsin y 2 p h p displaystyle theta arcsin y 2 pi h pi nbsp for some h Z displaystyle h in mathbb Z nbsp csc 8 r displaystyle csc theta r nbsp displaystyle iff nbsp 8 arccsc y 2 p h displaystyle theta operatorname arccsc y 2 pi h nbsp or 8 arccsc y 2 p h p displaystyle theta operatorname arccsc y 2 pi h pi nbsp for some h Z displaystyle h in mathbb Z nbsp cos 8 x displaystyle cos theta x nbsp displaystyle iff nbsp 8 arccos y 2 p k displaystyle theta arccos y 2 pi k nbsp or 8 arccos y 2 p k displaystyle theta arccos y 2 pi k nbsp for some k Z displaystyle k in mathbb Z nbsp sec 8 r displaystyle sec theta r nbsp displaystyle iff nbsp 8 arcsec y 2 p k displaystyle theta operatorname arcsec y 2 pi k nbsp or 8 arcsec y 2 p k displaystyle theta operatorname arcsec y 2 pi k nbsp for some k Z displaystyle k in mathbb Z nbsp For example if cos 8 1 displaystyle cos theta 1 nbsp then 8 p 2 p k p 2 p 1 k displaystyle theta pi 2 pi k pi 2 pi 1 k nbsp for some k Z displaystyle k in mathbb Z nbsp While if sin 8 1 displaystyle sin theta pm 1 nbsp then 8 p 2 p k p 2 p k 1 textstyle theta frac pi 2 pi k frac pi 2 pi k 1 nbsp for some k Z displaystyle k in mathbb Z nbsp where k displaystyle k nbsp will be even if sin 8 1 displaystyle sin theta 1 nbsp and it will be odd if sin 8 1 displaystyle sin theta 1 nbsp The equations sec 8 1 displaystyle sec theta 1 nbsp and csc 8 1 displaystyle csc theta pm 1 nbsp have the same solutions as cos 8 1 displaystyle cos theta 1 nbsp and sin 8 1 displaystyle sin theta pm 1 nbsp respectively In all equations above except for those just solved i e except for sin displaystyle sin nbsp csc 8 1 displaystyle csc theta pm 1 nbsp and cos displaystyle cos nbsp sec 8 1 displaystyle sec theta 1 nbsp the integer k displaystyle k nbsp in the solution s formula is uniquely determined by 8 displaystyle theta nbsp for fixed r s x displaystyle r s x nbsp and y displaystyle y nbsp Detailed example and explanation of the plus or minus symbol edit The solutions to cos 8 x displaystyle cos theta x nbsp and sec 8 x displaystyle sec theta x nbsp involve the plus or minus symbol displaystyle pm nbsp whose meaning is now clarified Only the solution to cos 8 x displaystyle cos theta x nbsp will be discussed since the discussion for sec 8 x displaystyle sec theta x nbsp is the same We are given x displaystyle x nbsp between 1 x 1 displaystyle 1 leq x leq 1 nbsp and we know that there is an angle 8 displaystyle theta nbsp in some interval that satisfies cos 8 x displaystyle cos theta x nbsp We want to find this 8 displaystyle theta nbsp The table above indicates that the solution is8 arccos x 2 p k for some k Z displaystyle theta pm arccos x 2 pi k quad text for some k in mathbb Z nbsp which is a shorthand way of saying that at least one of the following statement is true 8 arccos x 2 p k displaystyle theta arccos x 2 pi k nbsp for some integer k displaystyle k nbsp or 8 arccos x 2 p k displaystyle theta arccos x 2 pi k nbsp for some integer k displaystyle k nbsp As mentioned above if arccos x p displaystyle arccos x pi nbsp which by definition only happens when x cos p 1 displaystyle x cos pi 1 nbsp then both statements 1 and 2 hold although with different values for the integer k displaystyle k nbsp if K displaystyle K nbsp is the integer from statement 1 meaning that 8 p 2 p K displaystyle theta pi 2 pi K nbsp holds then the integer k displaystyle k nbsp for statement 2 is K 1 displaystyle K 1 nbsp because 8 p 2 p 1 K displaystyle theta pi 2 pi 1 K nbsp However if x 1 displaystyle x neq 1 nbsp then the integer k displaystyle k nbsp is unique and completely determined by 8 displaystyle theta nbsp If arccos x 0 displaystyle arccos x 0 nbsp which by definition only happens when x cos 0 1 displaystyle x cos 0 1 nbsp then arccos x 0 displaystyle pm arccos x 0 nbsp because arccos x 0 0 displaystyle arccos x 0 0 nbsp and arccos x 0 0 displaystyle arccos x 0 0 nbsp so in both cases arccos x displaystyle pm arccos x nbsp is equal to 0 displaystyle 0 nbsp 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 nbsp and arccos x p displaystyle arccos x pi nbsp we now focus on the case where arccos x 0 displaystyle arccos x neq 0 nbsp and arccos x p displaystyle arccos x neq pi nbsp So assume this from now on The solution to cos 8 x displaystyle cos theta x nbsp is still8 arccos x 2 p k for some k Z displaystyle theta pm arccos x 2 pi k quad text for some k in mathbb Z nbsp 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 nbsp and 0 lt arccos x lt p displaystyle 0 lt arccos x lt pi nbsp statements 1 and 2 are different and furthermore exactly one of the two equalities holds not both Additional information about 8 displaystyle theta nbsp is needed to determine which one holds For example suppose that x 0 displaystyle x 0 nbsp and that all that is known about 8 displaystyle theta nbsp is that p 8 p displaystyle pi leq theta leq pi nbsp and nothing more is known Then arccos x arccos 0 p 2 displaystyle arccos x arccos 0 frac pi 2 nbsp and moreover in this particular case k 0 displaystyle k 0 nbsp for both the displaystyle nbsp case and the displaystyle nbsp case and so consequently 8 arccos x 2 p k p 2 2 p 0 p 2 displaystyle theta pm arccos x 2 pi k pm left frac pi 2 right 2 pi 0 pm frac pi 2 nbsp This means that 8 displaystyle theta nbsp could be either p 2 displaystyle pi 2 nbsp or p 2 displaystyle pi 2 nbsp Without additional information it is not possible to determine which of these values 8 displaystyle theta nbsp has An example of some additional information that could determine the value of 8 displaystyle theta nbsp would be knowing that the angle is above the x displaystyle x nbsp axis in which case 8 p 2 displaystyle theta pi 2 nbsp or alternatively knowing that it is below the x displaystyle x nbsp axis in which case 8 p 2 displaystyle theta pi 2 nbsp Equal identical trigonometric functions edit The table below shows how two angles 8 displaystyle theta nbsp and f displaystyle varphi nbsp must be related if their values under a given trigonometric function are equal or negatives of each other Equation if and only if Solution for some k Z displaystyle k in mathbb Z nbsp Also a solution to sin 8 sin f displaystyle phantom sin theta sin varphi nbsp displaystyle iff nbsp 8 1 k f 2 p k p displaystyle theta phantom quad 1 k varphi phantom 2 pi k phantom pi nbsp csc 8 csc f displaystyle phantom csc theta csc varphi nbsp cos 8 cos f displaystyle phantom cos theta cos varphi nbsp displaystyle iff nbsp 8 1 f 2 p k p displaystyle theta phantom 1 quad pm varphi 2 pi k phantom pi nbsp sec 8 sec f displaystyle phantom sec theta sec varphi nbsp tan 8 tan f displaystyle phantom tan theta tan varphi nbsp displaystyle iff nbsp 8 1 k 1 f 2 p k p displaystyle theta phantom 1 k 1 varphi phantom 2 pi k phantom pi nbsp cot 8 cot f displaystyle phantom cot theta cot varphi nbsp sin 8 sin f displaystyle sin theta sin varphi nbsp displaystyle iff nbsp 8 1 k 1 f 2 p k p displaystyle theta 1 k 1 varphi phantom 2 pi k phantom pi nbsp csc 8 csc f displaystyle csc theta csc varphi nbsp cos 8 cos f displaystyle cos theta cos varphi nbsp displaystyle iff nbsp 8 1 f 2 p k p p displaystyle theta phantom 1 quad pm varphi 2 pi k pi phantom pi nbsp sec 8 sec f displaystyle sec theta sec varphi nbsp tan 8 tan f displaystyle tan theta tan varphi nbsp displaystyle iff nbsp 8 1 f 2 p k p displaystyle theta phantom 1 quad varphi phantom 2 pi k phantom pi nbsp cot 8 cot f displaystyle cot theta cot varphi nbsp sin 8 sin f cos 8 cos f displaystyle begin aligned phantom left sin theta right amp left sin varphi right amp Updownarrow phantom left cos theta right amp left cos varphi right end aligned nbsp displaystyle iff nbsp 8 1 f 2 p k p displaystyle theta phantom 1 quad pm varphi phantom 2 pi k phantom pi nbsp tan 8 tan f csc 8 csc f sec 8 sec f cot 8 cot f displaystyle begin aligned phantom left tan theta right amp left tan varphi right left csc theta right amp left csc varphi right left sec theta right amp left sec varphi right left cot theta right amp left cot varphi right end aligned nbsp The vertical double arrow displaystyle Updownarrow nbsp in the last row indicates that 8 displaystyle theta nbsp and f displaystyle varphi nbsp satisfy sin 8 sin f displaystyle left sin theta right left sin varphi right nbsp if and only if they satisfy cos 8 cos f displaystyle left cos theta right left cos varphi right nbsp Set of all solutions to elementary trigonometric equationsThus given a single solution 8 displaystyle theta nbsp to an elementary trigonometric equation sin 8 y displaystyle sin theta y nbsp is such an equation for instance and because sin arcsin y y displaystyle sin arcsin y y nbsp always holds 8 arcsin y displaystyle theta arcsin y nbsp is always a solution the set of all solutions to it are If 8 displaystyle theta nbsp solves then Set of all solutions in terms of 8 displaystyle theta nbsp sin 8 y displaystyle sin theta y nbsp then f sin f y displaystyle varphi sin varphi y nbsp 8 displaystyle theta nbsp 2 displaystyle 2 nbsp p Z displaystyle pi mathbb Z nbsp 8 displaystyle cup theta nbsp p displaystyle pi nbsp 2 p Z displaystyle 2 pi mathbb Z nbsp csc 8 r displaystyle csc theta r nbsp then f csc f r displaystyle varphi csc varphi r nbsp 8 displaystyle theta nbsp 2 displaystyle 2 nbsp p Z displaystyle pi mathbb Z nbsp 8 displaystyle cup theta nbsp p displaystyle pi nbsp 2 p Z displaystyle 2 pi mathbb Z nbsp cos 8 x displaystyle cos theta x nbsp then f cos f x displaystyle varphi cos varphi x nbsp 8 displaystyle theta nbsp 2 displaystyle 2 nbsp p Z displaystyle pi mathbb Z nbsp 8 displaystyle cup theta nbsp 2 p Z displaystyle 2 pi mathbb Z nbsp sec 8 r displaystyle sec theta r nbsp then f sec f r displaystyle varphi sec varphi r nbsp 8 displaystyle theta nbsp 2 displaystyle 2 nbsp p Z displaystyle pi mathbb Z nbsp 8 displaystyle cup theta nbsp 2 p Z displaystyle 2 pi mathbb Z nbsp tan 8 s displaystyle tan theta s nbsp then f tan f s displaystyle varphi tan varphi s nbsp 8 displaystyle theta nbsp displaystyle nbsp p Z displaystyle pi mathbb Z nbsp cot 8 r displaystyle cot theta r nbsp then f cot f r displaystyle varphi cot varphi r nbsp 8 displaystyle theta nbsp displaystyle nbsp p Z displaystyle pi mathbb Z, wikipedia, wiki , book, books, library,
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