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105-486: The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle . More modern definitions express the sine and cosine as infinite series , or as the solutions of certain differential equations , allowing their extension to arbitrary positive and negative values and even to complex numbers . The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves ,

210-426: A u {\displaystyle u} is called an upper bound of S . {\displaystyle S.} So, Dedekind completeness means that, if S is bounded above, it has an upper bound that is less than any other upper bound. Dedekind completeness implies other sorts of completeness (see below), but also has some important consequences. The last two properties are summarized by saying that

315-645: A = sin ⁡ β b = sin ⁡ γ c . {\displaystyle {\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{c}}.} This is equivalent to the equality of the first three expressions below: a sin ⁡ α = b sin ⁡ β = c sin ⁡ γ = 2 R , {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=2R,} where R {\displaystyle R}

420-430: A ⋅ b | a | | b | . {\displaystyle {\begin{aligned}\sin(\theta )&={\frac {|\mathbb {a} \times \mathbb {b} |}{|a||b|}},\\\cos(\theta )&={\frac {\mathbb {a} \cdot \mathbb {b} }{|a||b|}}.\end{aligned}}} The sine and cosine functions may also be defined in a more general way by using unit circle , a circle of radius one centered at

525-440: A , {\displaystyle D_{n-1}+a_{n}/10^{n}\leq a,} and one sets D n = D n − 1 + a n / 10 n . {\displaystyle D_{n}=D_{n-1}+a_{n}/10^{n}.} One can use the defining properties of the real numbers to show that x is the least upper bound of the D n . {\displaystyle D_{n}.} So,

630-480: A decimal point , representing the infinite series For example, for the circle constant π = 3.14159 ⋯ , {\displaystyle \pi =3.14159\cdots ,} k is zero and b 0 = 3 , {\displaystyle b_{0}=3,} a 1 = 1 , {\displaystyle a_{1}=1,} a 2 = 4 , {\displaystyle a_{2}=4,} etc. More formally,

735-406: A decimal representation for a nonnegative real number x consists of a nonnegative integer k and integers between zero and nine in the infinite sequence (If k > 0 , {\displaystyle k>0,} then by convention b k ≠ 0. {\displaystyle b_{k}\neq 0.} ) Such a decimal representation specifies the real number as

840-443: A line called the number line or real line , where the points corresponding to integers ( ..., −2, −1, 0, 1, 2, ... ) are equally spaced. Conversely, analytic geometry is the association of points on lines (especially axis lines ) to real numbers such that geometric displacements are proportional to differences between corresponding numbers. The informal descriptions above of the real numbers are not sufficient for ensuring

945-593: A power of ten , extending to finitely many positive powers of ten to the left and infinitely many negative powers of ten to the right. For a number x whose decimal representation extends k places to the left, the standard notation is the juxtaposition of the digits b k b k − 1 ⋯ b 0 . a 1 a 2 ⋯ , {\displaystyle b_{k}b_{k-1}\cdots b_{0}.a_{1}a_{2}\cdots ,} in descending order by power of ten, with non-negative and negative powers of ten separated by

1050-486: A total order that have the following properties. Many other properties can be deduced from the above ones. In particular: Several other operations are commonly used, which can be deduced from the above ones. The total order that is considered above is denoted a < b {\displaystyle a<b} and read as " a is less than b ". Three other order relations are also commonly used: The real numbers 0 and 1 are commonly identified with

1155-452: A characterization of the real numbers.) It is not true that R {\displaystyle \mathbb {R} } is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field , and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in

SECTION 10

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1260-507: A curve can be obtained by using the integral with a certain bounded interval. Their antiderivatives are: ∫ sin ⁡ ( x ) d x = − cos ⁡ ( x ) + C ∫ cos ⁡ ( x ) d x = sin ⁡ ( x ) + C , {\displaystyle \int \sin(x)\,dx=-\cos(x)+C\qquad \int \cos(x)\,dx=\sin(x)+C,} where C {\displaystyle C} denotes

1365-595: A degree N {\displaystyle N} —denoted as T ( x ) {\displaystyle T(x)} —is defined as: T ( x ) = a 0 + ∑ n = 1 N a n cos ⁡ ( n x ) + ∑ n = 1 N b n sin ⁡ ( n x ) . {\displaystyle T(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos(nx)+\sum _{n=1}^{N}b_{n}\sin(nx).} The trigonometric series can be defined similarly analogous to

1470-405: A limit, without computing it, and even without knowing it. For example, the standard series of the exponential function converges to a real number for every x , because the sums can be made arbitrarily small (independently of M ) by choosing N sufficiently large. This proves that the sequence is Cauchy, and thus converges, showing that e x {\displaystyle e^{x}}

1575-459: A nonnegative real number x , one can define a decimal representation of x by induction , as follows. Define b k ⋯ b 0 {\displaystyle b_{k}\cdots b_{0}} as decimal representation of the largest integer D 0 {\displaystyle D_{0}} such that D 0 ≤ x {\displaystyle D_{0}\leq x} (this integer exists because of

1680-483: A polynomial. Such a polynomial is known as the trigonometric polynomial . The trigonometric polynomial's ample applications may be acquired in its interpolation , and its extension of a periodic function known as the Fourier series . Let a n {\displaystyle a_{n}} and b n {\displaystyle b_{n}} be any coefficients, then the trigonometric polynomial of

1785-473: A rational number is an equivalence class of pairs of integers, and a real number is an equivalence class of Cauchy series), and are generally harmless. It is only in very specific situations, that one must avoid them and replace them by using explicitly the above homomorphisms. This is the case in constructive mathematics and computer programming . In the latter case, these homomorphisms are interpreted as type conversions that can often be done automatically by

1890-1422: A set of numbers composed of both real and imaginary numbers . For real number θ {\displaystyle \theta } , the definition of both sine and cosine functions can be extended in a complex plane in terms of an exponential function as follows: sin ⁡ ( θ ) = e i θ − e − i θ 2 i , cos ⁡ ( θ ) = e i θ + e − i θ 2 , {\displaystyle {\begin{aligned}\sin(\theta )&={\frac {e^{i\theta }-e^{-i\theta }}{2i}},\\\cos(\theta )&={\frac {e^{i\theta }+e^{-i\theta }}{2}},\end{aligned}}} Alternatively, both functions can be defined in terms of Euler's formula : e i θ = cos ⁡ ( θ ) + i sin ⁡ ( θ ) , e − i θ = cos ⁡ ( θ ) − i sin ⁡ ( θ ) . {\displaystyle {\begin{aligned}e^{i\theta }&=\cos(\theta )+i\sin(\theta ),\\e^{-i\theta }&=\cos(\theta )-i\sin(\theta ).\end{aligned}}} When plotted on

1995-480: Is consistent with the right-angled triangle definition of sine and cosine when 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}} because the length of the hypotenuse of the unit circle is always 1; mathematically speaking, the sine of an angle equals the opposite side of the triangle, which is simply the y {\displaystyle y} - coordinate. A similar argument can be made for

2100-708: Is cosine and that the derivative of cosine is the negative of sine. This means the successive derivatives of sin ⁡ ( x ) {\displaystyle \sin(x)} are cos ⁡ ( x ) {\displaystyle \cos(x)} , − sin ⁡ ( x ) {\displaystyle -\sin(x)} , − cos ⁡ ( x ) {\displaystyle -\cos(x)} , sin ⁡ ( x ) {\displaystyle \sin(x)} , continuing to repeat those four functions. The ( 4 n + k ) {\displaystyle (4n+k)} - th derivative, evaluated at

2205-401: Is decreasing (going downward)—in certain intervals. This information can be represented as a Cartesian coordinates system divided into four quadrants. Both sine and cosine functions can be defined by using differential equations. The pair of ( cos ⁡ θ , sin ⁡ θ ) {\displaystyle (\cos \theta ,\sin \theta )} is

SECTION 20

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2310-573: 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 . The standard range of principal values for arcsin is from − π 2 {\textstyle -{\frac {\pi }{2}}} to π 2 {\textstyle {\frac {\pi }{2}}} , and

2415-696: Is found in the functional equation for the Gamma function , which in turn is found in the functional equation for the Riemann zeta-function , As a holomorphic function , sin z is a 2D solution of Laplace's equation : The complex sine function is also related to the level curves of pendulums . The word sine is derived, indirectly, from the Sanskrit word jyā 'bow-string' or more specifically its synonym jīvá (both adopted from Ancient Greek χορδή 'string'), due to visual similarity between

2520-435: Is less than ε for all n and m that are both greater than N . This definition, originally provided by Cauchy , formalizes the fact that the x n eventually come and remain arbitrarily close to each other. A sequence ( x n ) converges to the limit x if its elements eventually come and remain arbitrarily close to x , that is, if for any ε > 0 there exists an integer N (possibly depending on ε) such that

2625-1053: Is said to be odd if f ( − x ) = − f ( x ) {\displaystyle f(-x)=-f(x)} , and is said to be even if f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} . The sine function is odd, whereas the cosine function is even. Both sine and cosine functions are similar, with their difference being shifted by π 2 {\textstyle {\frac {\pi }{2}}} . This means, sin ⁡ ( θ ) = cos ⁡ ( π 2 − θ ) , cos ⁡ ( θ ) = sin ⁡ ( π 2 − θ ) . {\displaystyle {\begin{aligned}\sin(\theta )&=\cos \left({\frac {\pi }{2}}-\theta \right),\\\cos(\theta )&=\sin \left({\frac {\pi }{2}}-\theta \right).\end{aligned}}} Zero

2730-487: Is so that many sequences have limits . More formally, the reals are complete (in the sense of metric spaces or uniform spaces , which is a different sense than the Dedekind completeness of the order in the previous section): A sequence ( x n ) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance | x n − x m |

2835-937: Is the gamma function and ϖ {\displaystyle \varpi } is the lemniscate constant . The inverse function of sine is arcsine or inverse sine, denoted as "arcsin", "asin", or sin − 1 {\displaystyle \sin ^{-1}} . The inverse function of cosine is arccosine, denoted as "arccos", "acos", or cos − 1 {\displaystyle \cos ^{-1}} . As sine and cosine are not injective , their inverses are not exact inverse functions, but partial inverse functions. 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} , and so on. It follows that

2940-787: Is the incomplete elliptic integral of the second kind with modulus k {\displaystyle k} . It cannot be expressed using elementary functions . In the case of a full period, its arc length is L = 4 2 π 3 Γ ( 1 / 4 ) 2 + Γ ( 1 / 4 ) 2 2 π = 2 π ϖ + 2 ϖ ≈ 7.6404 … {\displaystyle L={\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma (1/4)^{2}}}+{\frac {\Gamma (1/4)^{2}}{\sqrt {2\pi }}}={\frac {2\pi }{\varpi }}+2\varpi \approx 7.6404\ldots } where Γ {\displaystyle \Gamma }

3045-438: Is the angle between a {\displaystyle \mathbb {a} } and b {\displaystyle \mathbb {b} } , then sine and cosine can be defined as: sin ⁡ ( θ ) = | a × b | | a | | b | , cos ⁡ ( θ ) =

3150-546: Is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function is sin ⁡ ( 0 ) = 0 {\displaystyle \sin(0)=0} . The only real fixed point of the cosine function is called the Dottie number . The Dottie number is the unique real root of the equation cos ⁡ ( x ) = x {\displaystyle \cos(x)=x} . The decimal expansion of

3255-614: Is the triangle's circumradius . The law of cosines is useful for computing the length of an unknown side if two other sides and an angle are known. The law states, a 2 + b 2 − 2 a b cos ⁡ ( γ ) = c 2 {\displaystyle a^{2}+b^{2}-2ab\cos(\gamma )=c^{2}} In the case where γ = π / 2 {\displaystyle \gamma =\pi /2} from which cos ⁡ ( γ ) = 0 {\displaystyle \cos(\gamma )=0} ,

Sine and cosine - Misplaced Pages Continue

3360-458: Is well defined for every x . The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways. First, an order can be lattice-complete . It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z , z + 1 is larger). Additionally, an order can be Dedekind-complete, see § Axiomatic approach . The uniqueness result at

3465-495: Is what mathematicians and physicists did during several centuries before the first formal definitions were provided in the second half of the 19th century. See Construction of the real numbers for details about these formal definitions and the proof of their equivalence. The real numbers form an ordered field . Intuitively, this means that methods and rules of elementary arithmetic apply to them. More precisely, there are two binary operations , addition and multiplication , and

3570-648: The x {\displaystyle x} - axis. The x {\displaystyle x} - and y {\displaystyle y} - coordinates of this point of intersection are equal to cos ⁡ ( θ ) {\displaystyle \cos(\theta )} and sin ⁡ ( θ ) {\displaystyle \sin(\theta )} , respectively; that is, sin ⁡ ( θ ) = y , cos ⁡ ( θ ) = x . {\displaystyle \sin(\theta )=y,\qquad \cos(\theta )=x.} This definition

3675-565: The jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period ( Aryabhatiya and Surya Siddhanta ), via translation from Sanskrit to Arabic and then from Arabic to Latin. All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines , used in solving triangles . With the exception of the sine (which

3780-643: The compiler . Previous properties do not distinguish real numbers from rational numbers . This distinction is provided by Dedekind completeness , which states that every set of real numbers with an upper bound admits a least upper bound . This means the following. A set of real numbers S {\displaystyle S} is bounded above if there is a real number u {\displaystyle u} such that s ≤ u {\displaystyle s\leq u} for all s ∈ S {\displaystyle s\in S} ; such

3885-1748: The complex plane , the function e i x {\displaystyle e^{ix}} for real values of x {\displaystyle x} traces out the unit circle in the complex plane. Both sine and cosine functions may be simplified to the imaginary and real parts of e i θ {\displaystyle e^{i\theta }} as: sin ⁡ θ = Im ⁡ ( e i θ ) , cos ⁡ θ = Re ⁡ ( e i θ ) . {\displaystyle {\begin{aligned}\sin \theta &=\operatorname {Im} (e^{i\theta }),\\\cos \theta &=\operatorname {Re} (e^{i\theta }).\end{aligned}}} When z = x + i y {\displaystyle z=x+iy} for real values x {\displaystyle x} and y {\displaystyle y} , where i = − 1 {\displaystyle i={\sqrt {-1}}} , both sine and cosine functions can be expressed in terms of real sines, cosines, and hyperbolic functions as: sin ⁡ z = sin ⁡ x cosh ⁡ y + i cos ⁡ x sinh ⁡ y , cos ⁡ z = cos ⁡ x cosh ⁡ y − i sin ⁡ x sinh ⁡ y . {\displaystyle {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y,\\\cos z&=\cos x\cosh y-i\sin x\sinh y.\end{aligned}}} Sine and cosine are used to connect

3990-432: The concavity of a function can be defined by applying the inequality of the function's second derivative greater or less than equal to zero. The following table shows that both sine and cosine functions have concavity and monotonicity—the positive sign ( + {\displaystyle +} ) denotes a graph is increasing (going upward) and the negative sign ( − {\displaystyle -} )

4095-800: The constant of integration . These antiderivatives may be applied to compute the mensuration properties of both sine and cosine functions' curves with a given interval. For example, the arc length of the sine curve between 0 {\displaystyle 0} and t {\displaystyle t} is ∫ 0 t 1 + cos 2 ⁡ ( x ) d x = 2 E ⁡ ( t , 1 2 ) , {\displaystyle \int _{0}^{t}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}\operatorname {E} \left(t,{\frac {1}{\sqrt {2}}}\right),} where E ⁡ ( φ , k ) {\displaystyle \operatorname {E} (\varphi ,k)}

4200-1788: The hyperbolic sine and cosine . These are entire functions . It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of its argument: Using the partial fraction expansion technique in complex analysis , one can find that the infinite series ∑ n = − ∞ ∞ ( − 1 ) n z − n = 1 z − 2 z ∑ n = 1 ∞ ( − 1 ) n n 2 − z 2 {\displaystyle \sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{z-n}}={\frac {1}{z}}-2z\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{2}-z^{2}}}} both converge and are equal to π sin ⁡ ( π z ) {\textstyle {\frac {\pi }{\sin(\pi z)}}} . Similarly, one can show that π 2 sin 2 ⁡ ( π z ) = ∑ n = − ∞ ∞ 1 ( z − n ) 2 . {\displaystyle {\frac {\pi ^{2}}{\sin ^{2}(\pi z)}}=\sum _{n=-\infty }^{\infty }{\frac {1}{(z-n)^{2}}}.} Using product expansion technique, one can derive sin ⁡ ( π z ) = π z ∏ n = 1 ∞ ( 1 − z 2 n 2 ) . {\displaystyle \sin(\pi z)=\pi z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right).} sin( z )

4305-400: The initial conditions y ( 0 ) = 0 {\displaystyle y(0)=0} and x ( 0 ) = 1 {\displaystyle x(0)=1} . One could interpret the unit circle in the above definitions as defining the phase space trajectory of the differential equation with the given initial conditions. It can be interpreted as a phase space trajectory of

Sine and cosine - Misplaced Pages Continue

4410-422: The natural numbers 0 and 1 . This allows identifying any natural number n with the sum of n real numbers equal to 1 . This identification can be pursued by identifying a negative integer − n {\displaystyle -n} (where n {\displaystyle n} is a natural number) with the additive inverse − n {\displaystyle -n} of

4515-570: The square roots of −1 . The real numbers include the rational numbers , such as the integer −5 and the fraction 4 / 3 . The rest of the real numbers are called irrational numbers . Some irrational numbers (as well as all the rationals) are the root of a polynomial with integer coefficients, such as the square root √2 = 1.414... ; these are called algebraic numbers . There are also real numbers which are not, such as π = 3.1415... ; these are called transcendental numbers . Real numbers can be thought of as all points on

4620-501: The 12th century by Gerard of Cremona , he used the Latin equivalent sinus (which also means 'bay' or 'fold', and more specifically 'the hanging fold of a toga over the breast'). Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage. The English form sine was introduced in the 1590s. The word cosine derives from an abbreviation of

4725-511: The Archimedean property). Then, supposing by induction that the decimal fraction D i {\displaystyle D_{i}} has been defined for i < n , {\displaystyle i<n,} one defines a n {\displaystyle a_{n}} as the largest digit such that D n − 1 + a n / 10 n ≤

4830-590: The Dottie number is approximately 0.739085. The sine and cosine functions are infinitely differentiable. The derivative of sine is cosine, and the derivative of cosine is negative sine: d d x sin ⁡ ( x ) = cos ⁡ ( x ) , d d x cos ⁡ ( x ) = − sin ⁡ ( x ) . {\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x),\qquad {\frac {d}{dx}}\cos(x)=-\sin(x).} Continuing

4935-583: The Latin complementi sinus 'sine of the complementary angle ' as cosinus in Edmund Gunter 's Canon triangulorum (1620), which also includes a similar definition of cotangens . While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The sine and cosine functions can be traced to

5040-482: The abbreviations sin , cos , and tan is by the 16th-century French mathematician Albert Girard ; these were further promulgated by Euler (see below). The Opus palatinum de triangulis of Georg Joachim Rheticus , a student of Copernicus , was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work

5145-407: The accompanying figure, angle α {\displaystyle \alpha } in a right triangle A B C {\displaystyle ABC} is the angle of interest. The three sides of the triangle are named as follows: Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse, and the cosine of

5250-1361: The adjacent side. The cotangent function is the ratio between the adjacent and opposite sides, a reciprocal of a tangent function. These functions can be formulated as: tan ⁡ ( θ ) = sin ⁡ ( θ ) cos ⁡ ( θ ) = opposite adjacent , cot ⁡ ( θ ) = 1 tan ⁡ ( θ ) = adjacent opposite , csc ⁡ ( θ ) = 1 sin ⁡ ( θ ) = hypotenuse opposite , sec ⁡ ( θ ) = 1 cos ⁡ ( θ ) = hypotenuse adjacent . {\displaystyle {\begin{aligned}\tan(\theta )&={\frac {\sin(\theta )}{\cos(\theta )}}={\frac {\text{opposite}}{\text{adjacent}}},\\\cot(\theta )&={\frac {1}{\tan(\theta )}}={\frac {\text{adjacent}}{\text{opposite}}},\\\csc(\theta )&={\frac {1}{\sin(\theta )}}={\frac {\text{hypotenuse}}{\text{opposite}}},\\\sec(\theta )&={\frac {1}{\cos(\theta )}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}.\end{aligned}}} As stated,

5355-406: The angle can be defined similarly; for example, the tangent is the ratio between the opposite and adjacent sides or equivalently the ratio between the sine and cosine functions. The reciprocal of sine is cosecant, which gives the ratio of the hypotenuse length to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the ratio of the hypotenuse length to that of

SECTION 50

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5460-473: The angle is equal to the length of the adjacent side divided by the length of the hypotenuse: sin ⁡ ( α ) = opposite hypotenuse , cos ⁡ ( α ) = adjacent hypotenuse . {\displaystyle \sin(\alpha )={\frac {\text{opposite}}{\text{hypotenuse}}},\qquad \cos(\alpha )={\frac {\text{adjacent}}{\text{hypotenuse}}}.} The other trigonometric functions of

5565-588: The arc of a circle with its corresponding chord and a bow with its string (see jyā, koti-jyā and utkrama-jyā ). This was transliterated in Arabic as jība , which is meaningless in that language and written as jb ( جب ). Since Arabic is written without short vowels, jb was interpreted as the homograph jayb ( جيب ), which means 'bosom', 'pocket', or 'fold'. When the Arabic texts of Al-Battani and al-Khwārizmī were translated into Medieval Latin in

5670-407: The arcsine function is multivalued: arcsin ⁡ ( 0 ) = 0 {\displaystyle \arcsin(0)=0} , but also arcsin ⁡ ( 0 ) = π {\displaystyle \arcsin(0)=\pi } , arcsin ⁡ ( 0 ) = 2 π {\displaystyle \arcsin(0)=2\pi } , and so on. When only one value

5775-529: The axiomatic definition and are thus equivalent. Real numbers are completely characterized by their fundamental properties that can be summarized by saying that they form an ordered field that is Dedekind complete . Here, "completely characterized" means that there is a unique isomorphism between any two Dedekind complete ordered fields, and thus that their elements have exactly the same properties. This implies that one can manipulate real numbers and compute with them, without knowing how they can be defined; this

5880-441: The axioms of Zermelo–Fraenkel set theory including the axiom of choice (ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it. As a topological space, the real numbers are separable . This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have

5985-420: The cardinality of the power set of the set of the natural numbers. The statement that there is no subset of the reals with cardinality strictly greater than ℵ 0 {\displaystyle \aleph _{0}} and strictly smaller than c {\displaystyle {\mathfrak {c}}} is known as the continuum hypothesis (CH). It is neither provable nor refutable using

6090-808: The case of a Fourier series with a given integrable function f {\displaystyle f} , the coefficients of a trigonometric series are: A n = 1 π ∫ 0 2 π f ( x ) cos ⁡ ( n x ) d x , B n = 1 π ∫ 0 2 π f ( x ) sin ⁡ ( n x ) d x . {\displaystyle {\begin{aligned}A_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\cos(nx)\,dx,\\B_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\sin(nx)\,dx.\end{aligned}}} Both sine and cosine can be extended further via complex number ,

6195-480: The circumference of a circle, depending on the input θ > 0 {\displaystyle \theta >0} . In a sine function, if the input is θ = π 2 {\textstyle \theta ={\frac {\pi }{2}}} , the point is rotated counterclockwise and stopped exactly on the y {\displaystyle y} - axis. If θ = π {\displaystyle \theta =\pi } ,

6300-411: The classical definitions of limits , continuity and derivatives . The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold R , often using blackboard bold , ⁠ R {\displaystyle \mathbb {R} } ⁠ . The adjective real , used in the 17th century by René Descartes , distinguishes real numbers from imaginary numbers such as

6405-439: The construction of the reals from surreal numbers , since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield. The set of all real numbers is uncountable , in the sense that while both the set of all natural numbers {1, 2, 3, 4, ...} and the set of all real numbers are infinite sets , there exists no one-to-one function from

SECTION 60

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6510-652: The correctness of proofs of theorems involving real numbers. The realization that a better definition was needed, and the elaboration of such a definition was a major development of 19th-century mathematics and is the foundation of real analysis , the study of real functions and real-valued sequences . A current axiomatic definition is that real numbers form the unique ( up to an isomorphism ) Dedekind-complete ordered field . Other common definitions of real numbers include equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts , and infinite decimal representations . All these definitions satisfy

6615-695: The cosine function as well, although the point is rotated initially from the y {\displaystyle y} - coordinate. In other words, both sine and cosine functions are periodic , meaning any angle added by the circumference's circle is the angle itself. Mathematically, sin ⁡ ( θ + 2 π ) = sin ⁡ ( θ ) , cos ⁡ ( θ + 2 π ) = cos ⁡ ( θ ) . {\displaystyle \sin(\theta +2\pi )=\sin(\theta ),\qquad \cos(\theta +2\pi )=\cos(\theta ).} A function f {\displaystyle f}

6720-406: The cosine function to show that the cosine of an angle when 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}} , even under the new definition using the unit circle. Using the unit circle definition has the advantage of drawing a graph of sine and cosine functions. This can be done by rotating counterclockwise a point along

6825-851: The derivative of each term gives the Taylor series for cosine: cos ⁡ ( x ) = 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + ⋯ = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! x 2 n {\displaystyle {\begin{aligned}\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\end{aligned}}} Both sine and cosine functions with multiple angles may appear as their linear combination , resulting in

6930-417: The distance | x n − x | is less than ε for n greater than N . Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete. The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of

7035-496: The end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way. These two notions of completeness ignore

7140-970: The equations: sin ⁡ ( arcsin ⁡ ( x ) ) = x cos ⁡ ( arccos ⁡ ( x ) ) = x {\displaystyle \sin(\arcsin(x))=x\qquad \cos(\arccos(x))=x} and arcsin ⁡ ( sin ⁡ ( θ ) ) = θ for − π 2 ≤ θ ≤ π 2 arccos ⁡ ( cos ⁡ ( θ ) ) = θ for 0 ≤ θ ≤ π {\displaystyle {\begin{aligned}\arcsin(\sin(\theta ))=\theta \quad &{\text{for}}\quad -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{for}}\quad 0\leq \theta \leq \pi \end{aligned}}} According to Pythagorean theorem ,

7245-427: The field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness ; the description in § Completeness is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces , since the definition of metric space relies on already having

7350-800: The first decimal representation, all a n {\displaystyle a_{n}} are zero for n > h , {\displaystyle n>h,} and, in the second representation, all a n {\displaystyle a_{n}} 9. (see 0.999... for details). In summary, there is a bijection between the real numbers and the decimal representations that do not end with infinitely many trailing 9. The preceding considerations apply directly for every numeral base B ≥ 2 , {\displaystyle B\geq 2,} simply by replacing 10 with B {\displaystyle B} and 9 with B − 1. {\displaystyle B-1.} A main reason for using real numbers

7455-1441: The following double-angle formulas: sin ⁡ ( 2 θ ) = 2 sin ⁡ ( θ ) cos ⁡ ( θ ) , cos ⁡ ( 2 θ ) = cos 2 ⁡ ( θ ) − sin 2 ⁡ ( θ ) = 2 cos 2 ⁡ ( θ ) − 1 = 1 − 2 sin 2 ⁡ ( θ ) {\displaystyle {\begin{aligned}\sin(2\theta )&=2\sin(\theta )\cos(\theta ),\\\cos(2\theta )&=\cos ^{2}(\theta )-\sin ^{2}(\theta )\\&=2\cos ^{2}(\theta )-1\\&=1-2\sin ^{2}(\theta )\end{aligned}}} The cosine double angle formula implies that sin and cos are, themselves, shifted and scaled sine waves. Specifically, sin 2 ⁡ ( θ ) = 1 − cos ⁡ ( 2 θ ) 2 cos 2 ⁡ ( θ ) = 1 + cos ⁡ ( 2 θ ) 2 {\displaystyle \sin ^{2}(\theta )={\frac {1-\cos(2\theta )}{2}}\qquad \cos ^{2}(\theta )={\frac {1+\cos(2\theta )}{2}}} The graph shows both sine and sine squared functions, with

7560-556: The identification of natural numbers with some real numbers is justified by the fact that Peano axioms are satisfied by these real numbers, with the addition with 1 taken as the successor function . Formally, one has an injective homomorphism of ordered monoids from the natural numbers N {\displaystyle \mathbb {N} } to the integers Z , {\displaystyle \mathbb {Z} ,} an injective homomorphism of ordered rings from Z {\displaystyle \mathbb {Z} } to

7665-408: The least upper bound of the decimal fractions that are obtained by truncating the sequence: given a positive integer n , the truncation of the sequence at the place n is the finite partial sum The real number x defined by the sequence is the least upper bound of the D n , {\displaystyle D_{n},} which exists by Dedekind completeness. Conversely, given

7770-599: The lengths of the unknown sides in a triangle if two angles and one side are known. Given that a triangle A B C {\displaystyle ABC} with sides a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} , and angles opposite those sides α {\displaystyle \alpha } , β {\displaystyle \beta } , and γ {\displaystyle \gamma } . The law states, sin ⁡ α

7875-450: The magnitude and angle of the complex number z {\displaystyle z} . For any real number θ {\displaystyle \theta } , Euler's formula in terms of polar coordinates is stated as z = r e i θ {\textstyle z=re^{i\theta }} . Applying the series definition of the sine and cosine to a complex argument, z , gives: where sinh and cosh are

7980-609: The metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals form a contractible (hence connected and simply connected ), separable and complete metric space of Hausdorff dimension  1. The real numbers are locally compact but not compact . There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to

8085-471: The origin ( 0 , 0 ) {\displaystyle (0,0)} , formulated as the equation of x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} in the Cartesian coordinate system . Let a line through the origin intersect the unit circle, making an angle of θ {\displaystyle \theta } with the positive half of

8190-464: The phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way. But the original use of the phrase "complete Archimedean field" was by David Hilbert , who meant still something else by it. He meant that

8295-512: The point 0: sin ( 4 n + k ) ⁡ ( 0 ) = { 0 when  k = 0 1 when  k = 1 0 when  k = 2 − 1 when  k = 3 {\displaystyle \sin ^{(4n+k)}(0)={\begin{cases}0&{\text{when }}k=0\\1&{\text{when }}k=1\\0&{\text{when }}k=2\\-1&{\text{when }}k=3\end{cases}}} where

8400-469: The point is at the circle's halfway. If θ = 2 π {\displaystyle \theta =2\pi } , the point returned to its origin. This results that both sine and cosine functions have the range between − 1 ≤ y ≤ 1 {\displaystyle -1\leq y\leq 1} . Extending the angle to any real domain, the point rotated counterclockwise continuously. This can be done similarly for

8505-499: The position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. They can be traced to the jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period . To define the sine and cosine of an acute angle α {\displaystyle \alpha } , start with a right triangle that contains an angle of measure α {\displaystyle \alpha } ; in

8610-399: The positive square root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive square root of 2). The completeness property of the reals is the basis on which calculus , and more generally mathematical analysis , are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has

8715-418: The process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself. These derivatives can be applied to the first derivative test , according to which the monotonicity of a function can be defined as the inequality of function's first derivative greater or less than equal to zero. It can also be applied to second derivative test , according to which

8820-492: The rational numbers Q , {\displaystyle \mathbb {Q} ,} and an injective homomorphism of ordered fields from Q {\displaystyle \mathbb {Q} } to the real numbers R . {\displaystyle \mathbb {R} .} The identifications consist of not distinguishing the source and the image of each injective homomorphism, and thus to write These identifications are formally abuses of notation (since, formally,

8925-533: The rational numbers an ordered subfield of the real numbers R . {\displaystyle \mathbb {R} .} The Dedekind completeness described below implies that some real numbers, such as 2 , {\displaystyle {\sqrt {2}},} are not rational numbers; they are called irrational numbers . The above identifications make sense, since natural numbers, integers and real numbers are generally not defined by their individual nature, but by defining properties ( axioms ). So,

9030-451: The ratios are the same for each of them. For example, each leg of the 45-45-90 right triangle is 1 unit, and its hypotenuse is 2 {\displaystyle {\sqrt {2}}} ; therefore, sin ⁡ 45 ∘ = cos ⁡ 45 ∘ = 2 2 {\textstyle \sin 45^{\circ }=\cos 45^{\circ }={\frac {\sqrt {2}}{2}}} . The following table shows

9135-907: The real and imaginary parts of a complex number with its polar coordinates ( r , θ ) {\displaystyle (r,\theta )} : z = r ( cos ⁡ ( θ ) + i sin ⁡ ( θ ) ) , {\displaystyle z=r(\cos(\theta )+i\sin(\theta )),} and the real and imaginary parts are Re ⁡ ( z ) = r cos ⁡ ( θ ) , Im ⁡ ( z ) = r sin ⁡ ( θ ) , {\displaystyle {\begin{aligned}\operatorname {Re} (z)&=r\cos(\theta ),\\\operatorname {Im} (z)&=r\sin(\theta ),\end{aligned}}} where r {\displaystyle r} and θ {\displaystyle \theta } represent

9240-464: The real number identified with n . {\displaystyle n.} Similarly a rational number p / q {\displaystyle p/q} (where p and q are integers and q ≠ 0 {\displaystyle q\neq 0} ) is identified with the division of the real numbers identified with p and q . These identifications make the set Q {\displaystyle \mathbb {Q} } of

9345-436: The real numbers form a real closed field . This implies the real version of the fundamental theorem of algebra , namely that every polynomial with real coefficients can be factored into polynomials with real coefficients of degree at most two. The most common way of describing a real number is via its decimal representation , a sequence of decimal digits each representing the product of an integer between zero and nine times

9450-417: The real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R {\displaystyle \mathbb {R} } . Thus R {\displaystyle \mathbb {R} } is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to

9555-429: The real numbers to the natural numbers. The cardinality of the set of all real numbers is denoted by c . {\displaystyle {\mathfrak {c}}.} and called the cardinality of the continuum . It is strictly greater than the cardinality of the set of all natural numbers (denoted ℵ 0 {\displaystyle \aleph _{0}} and called 'aleph-naught' ), and equals

9660-563: The resulting equation becomes the Pythagorean theorem . The cross product and dot product are operations on two vectors in Euclidean vector space . The sine and cosine functions can be defined in terms of the cross product and dot product. If a {\displaystyle \mathbb {a} } and b {\displaystyle \mathbb {b} } are vectors, and θ {\displaystyle \theta }

9765-496: The resulting sequence of digits is called a decimal representation of x . Another decimal representation can be obtained by replacing ≤ x {\displaystyle \leq x} with < x {\displaystyle <x} in the preceding construction. These two representations are identical, unless x is a decimal fraction of the form m 10 h . {\textstyle {\frac {m}{10^{h}}}.} In this case, in

9870-425: The same cardinality as the reals. The real numbers form a metric space : the distance between x and y is defined as the absolute value | x − y | . By virtue of being a totally ordered set, they also carry an order topology ; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in

9975-423: The sine in blue and the sine squared in red. Both graphs have the same shape but with different ranges of values and different periods. Sine squared has only positive values, but twice the number of periods. Both sine and cosine functions can be defined by using a Taylor series , a power series involving the higher-order derivatives. As mentioned in § Continuity and differentiation , the derivative of sine

10080-505: The solution ( x ( θ ) , y ( θ ) ) {\displaystyle (x(\theta ),y(\theta ))} to the two-dimensional system of differential equations y ′ ( θ ) = x ( θ ) {\displaystyle y'(\theta )=x(\theta )} and x ′ ( θ ) = − y ( θ ) {\displaystyle x'(\theta )=-y(\theta )} with

10185-417: The special value of each input for both sine and cosine with the domain between 0 < α < π 2 {\textstyle 0<\alpha <{\frac {\pi }{2}}} . The input in this table provides various unit systems such as degree, radian, and so on. The angles other than those five can be obtained by using a calculator. The law of sines is useful for computing

10290-478: The squared hypotenuse is the sum of two squared legs of a right triangle. Dividing the formula on both sides with squared hypotenuse resulting in the Pythagorean trigonometric identity , the sum of a squared sine and a squared cosine equals 1: sin 2 ⁡ ( θ ) + cos 2 ⁡ ( θ ) = 1. {\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1.} Sine and cosine satisfy

10395-1470: The standard range for arccos is from 0 {\displaystyle 0} to π {\displaystyle \pi } . The inverse function of both sine and cosine are defined as: θ = arcsin ⁡ ( opposite hypotenuse ) = arccos ⁡ ( adjacent hypotenuse ) , {\displaystyle \theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\right),} where for some integer k {\displaystyle k} , sin ⁡ ( y ) = x ⟺ y = arcsin ⁡ ( x ) + 2 π k ,  or  y = π − arcsin ⁡ ( x ) + 2 π k cos ⁡ ( y ) = x ⟺ y = arccos ⁡ ( x ) + 2 π k ,  or  y = − arccos ⁡ ( x ) + 2 π k {\displaystyle {\begin{aligned}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ or }}\\&y=\pi -\arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\arccos(x)+2\pi k,{\text{ or }}\\&y=-\arccos(x)+2\pi k\end{aligned}}} By definition, both functions satisfy

10500-1107: The superscript represents repeated differentiation. This implies the following Taylor series expansion at x = 0 {\displaystyle x=0} . One can then use the theory of Taylor series to show that the following identities hold for all real numbers x {\displaystyle x} —where x {\displaystyle x} is the angle in radians. More generally, for all complex numbers : sin ⁡ ( x ) = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) ! x 2 n + 1 {\displaystyle {\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\end{aligned}}} Taking

10605-545: The system of differential equations y ′ ( θ ) = x ( θ ) {\displaystyle y'(\theta )=x(\theta )} and x ′ ( θ ) = − y ( θ ) {\displaystyle x'(\theta )=-y(\theta )} starting from the initial conditions y ( 0 ) = 0 {\displaystyle y(0)=0} and x ( 0 ) = 1 {\displaystyle x(0)=1} . Their area under

10710-588: The trigonometric polynomial, its infinite inversion. Let A n {\displaystyle A_{n}} and B n {\displaystyle B_{n}} be any coefficients, then the trigonometric series can be defined as: 1 2 A 0 + ∑ n = 1 ∞ A n cos ⁡ ( n x ) + B n sin ⁡ ( n x ) . {\displaystyle {\frac {1}{2}}A_{0}+\sum _{n=1}^{\infty }A_{n}\cos(nx)+B_{n}\sin(nx).} In

10815-423: The values sin ⁡ ( α ) {\displaystyle \sin(\alpha )} and cos ⁡ ( α ) {\displaystyle \cos(\alpha )} appear to depend on the choice of a right triangle containing an angle of measure α {\displaystyle \alpha } . However, this is not the case as all such triangles are similar , and so

10920-480: Was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Arabic mathematicians, including the cosine, tangent, cotangent, secant and cosecant. Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents. Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. The first published use of

11025-607: Was finished by Rheticus' student Valentin Otho in 1596. Real number In mathematics , a real number is a number that can be used to measure a continuous one- dimensional quantity such as a distance , duration or temperature . Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion . The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in

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