Inverse functions of sin, cos, tan, etc.
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 .
Notation [ edit ] 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 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.[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.
Basic concepts [ edit ] 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 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[note 1] for more details and 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 ( r ) + 2 π h {\displaystyle \theta =\;\;\;\,\operatorname {arccsc}(r)+2\pi h} or θ = − arccsc ( r ) + 2 π h + π {\displaystyle \theta =-\operatorname {arccsc}(r)+2\pi h+\pi } for some h ∈ Z {\displaystyle h\in \mathbb {Z} } cos θ = x {\displaystyle \cos \theta =x} ⟺ {\displaystyle \iff } θ = arccos ( x ) + 2 π k {\displaystyle \theta =\;\;\;\,\arccos(x)+2\pi k} or θ = − arccos ( x ) + 2 π k {\displaystyle \theta =-\arccos(x)+2\pi k} for some k ∈ Z {\displaystyle k\in \mathbb {Z} } sec θ = r {\displaystyle \sec \theta =r} ⟺ {\displaystyle \iff } θ = arcsec ( r ) + 2 π k {\displaystyle \theta =\;\;\;\,\operatorname {arcsec}(r)+2\pi k} or θ = − arcsec ( r ) + 2 π k {\displaystyle \theta =-\operatorname {arcsec}(r)+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} ).
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 ± [ 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} }
Transforming equations [ edit ] The equations above can be transformed by using the reflection and shift identities:[18]
Transforming equations by shifts and reflections Argument: _ = {\displaystyle {\underline {\;~~~~~~\;}}=} − θ {\displaystyle -\theta } π 2 ± θ {\displaystyle {\frac {\pi }{2}}\pm \theta } π ± θ {\displaystyle \pi \pm \theta } 3 π 2 ± θ {\displaystyle {\frac {3\pi }{2}}\pm \theta } 2 k π ± θ , {\displaystyle 2k\pi \pm \theta ,} ( k ∈ Z ) {\displaystyle (k\in \mathbb {Z} )} sin _ = {\displaystyle \sin {\underline {\;~~~~~~~~~~~~~~\;}}=} − sin θ {\displaystyle -\sin \theta } − cos θ {\displaystyle {\phantom {-}}\cos \theta } ∓ sin θ {\displaystyle \mp \sin \theta } − cos θ {\displaystyle -\cos \theta } ± sin θ {\displaystyle \pm \sin \theta } csc _ = {\displaystyle \csc {\underline {\;~~~~~~~~~~~~~~\;}}=} − csc θ {\displaystyle -\csc \theta } − sec θ {\displaystyle {\phantom {-}}\sec \theta } ∓ csc θ {\displaystyle \mp \csc \theta } − sec θ {\displaystyle -\sec \theta } ± csc θ {\displaystyle \pm \csc \theta } cos _ = {\displaystyle \cos {\underline {\;~~~~~~~~~~~~~~\;}}=} − cos θ {\displaystyle {\phantom {-}}\cos \theta } ∓ sin θ {\displaystyle \mp \sin \theta } − cos θ {\displaystyle -\cos \theta } ± sin θ {\displaystyle \pm \sin \theta } − cos θ {\displaystyle {\phantom {-}}\cos \theta } sec _ = {\displaystyle \sec {\underline {\;~~~~~~~~~~~~~~\;}}=} − sec θ {\displaystyle {\phantom {-}}\sec \theta } ∓ csc θ {\displaystyle \mp \csc \theta } − sec θ {\displaystyle -\sec \theta } ± csc θ {\displaystyle \pm \csc \theta } − sec θ {\displaystyle {\phantom {-}}\sec \theta } tan _ = {\displaystyle \tan {\underline {\;~~~~~~~~~~~~~~\;}}=} − tan θ {\displaystyle -\tan \theta } ∓ cot θ {\displaystyle \mp \cot \theta } ± tan θ {\displaystyle \pm \tan \theta } ∓ cot θ {\displaystyle \mp \cot \theta } ± tan θ {\displaystyle \pm \tan \theta } cot _ = {\displaystyle \cot {\underline {\;~~~~~~~~~~~~~~\;}}=} − cot θ {\displaystyle -\cot \theta } ∓ tan θ {\displaystyle \mp \tan \theta } ± cot θ {\displaystyle \pm \cot \theta } ∓ tan θ {\displaystyle \mp \tan \theta } ± cot θ {\displaystyle \pm \cot \theta }
These formulas imply, in particular, that the following hold:
sin θ = − sin ( − θ ) = − sin ( π + θ ) = − sin ( π − θ ) = − cos ( π 2 + θ ) = − cos ( π 2 − θ ) = − cos ( − π 2 − θ ) = − cos ( − π 2 + θ ) = − cos ( 3 π 2 − θ ) = − cos ( − 3 π 2 + θ ) cos θ = − cos ( − θ ) = − cos ( π + θ ) = − cos ( π − θ ) = − sin ( π 2 + θ ) = − sin ( π 2 − θ ) = − sin ( − π 2 − θ ) = − sin ( − π 2 + θ ) = − sin ( 3 π 2 − θ ) = − sin ( − 3 π 2 + θ ) tan θ = − tan ( − θ ) = − tan ( π + θ ) = − tan ( π − θ ) = − cot ( π 2 + θ ) = − cot ( π 2 − θ ) = − cot ( − π 2 − θ ) = − cot ( − π 2 + θ ) = − cot ( 3 π 2 − θ ) = − cot ( − 3 π 2 + θ ) {\displaystyle {\begin{aligned}\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 )&&={\phantom {-}}\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{aligned}}} where swapping sin ↔ csc , {\displaystyle \sin \leftrightarrow \csc ,}