Trigonometric functions: Difference between revisions

In [[mathematics]], the '''trigonometric functions''' (also called '''circular functions''', '''angle functions''' or '''goniometric functions''')<ref name="Klein_1924"/><ref name="Klein_2004"/>{{r|klein}} are [[real function]]s which relate an angle of a [[right-angled triangle]] to ratios of two side lengths. They are widely used in all sciences that are related to [[geometry]], such as [[navigation]], [[solid mechanics]], [[celestial mechanics]], [[geodesy]], and many others. They are among the simplest [[periodic function]]s, and as such are also widely used for studying periodic phenomena through [[Fourier analysis]].

[[File:TrigFunctionDiagram.svg|thumb|Plot of the six trigonometric functions, the unit circle, and a line for the angle {{math|1=''θ'' = 0.7 radians}}. The points labelledlabeled {{color|#D00|1}}, {{color|#02D|Sec(''θ'')}}, {{color|#0D1|Csc(''θ'')}} represent the length of the line segment from the origin to that point. {{color|#D00|Sin(''θ'')}}, {{color|#02D|Tan(''θ'')}}, and {{color|#0D1|1}} are the heights to the line starting from the {{mvar|x}}-axis, while {{color|#D00|Cos(''θ'')}}, {{color|#02D|1}}, and {{color|#0D1|Cot(''θ'')}} are lengths along the {{mvar|x}}-axis starting from the origin.]]

In a right-angled triangle, the sum of the two acute angles is a right angle, that is, {{math|90°}} or {{math|{{sfrac|π|2}} [[radian]]s}}. Therefore <math>\sin(\theta)</math> and <math>\cos(90^\circ - \theta)</math> represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.

[[File:Periodic sine.svg|thumb|'''Top:''' Trigonometric function {{math|sin ''θ''}} for selected angles {{math|''θ''}}, {{math|{{pi}} − ''θ''}}, {{math|{{pi}} + ''θ''}}, and {{math|2{{pi}} − ''θ''}} in the four quadrants.<br>'''Bottom:''' Graph of sine function versus angle. Angles from the top panel are identified.]]

However, in [[calculus]] and [[mathematical analysis]], the trigonometric functions are generally regarded more abstractly as functions of [[real number|real]] or [[complex number]]s, rather than angles. In fact, the functions {{math|sin}} and {{math|cos}} can be defined for all complex numbers in terms of the [[exponential function]], via power series,<ref name=":0">{{Cite book|last=Rudin, Walter, 1921–2010|url=https://www.worldcat.org/oclc/1502474|title=Principles of mathematical analysis|isbn=0-07-054235-X|edition=Third |location=New York|oclc=1502474}}</ref> or as solutions to [[differential equation]]s given particular initial values<ref>{{Cite journal|last=Diamond|first=Harvey|date=2014|title=Defining Exponential and Trigonometric Functions Using Differential Equations|url=https://www.tandfonline.com/doi/full/10.4169/math.mag.87.1.37|journal=Mathematics Magazine|language=en|volume=87|issue=1|pages=37–42|doi=10.4169/math.mag.87.1.37|s2cid=126217060|issn=0025-570X}}</ref> (''see below''), without reference to any geometric notions. The other four trigonometric functions ({{math|tan}}, {{math|cot}}, {{math|sec}}, {{math|csc}}) can be defined as quotients and reciprocals of {{math|sin}} and {{math|cos}}, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions ''if'' ''the argument is regarded as an angle given in radians''.<ref name=":0" /> Moreover, these definitions result in simple expressions for the [[derivative]]s and [[Antiderivative|indefinite integrals]] for the trigonometric functions.<ref name=":1">{{Cite book|last=Spivak|first=Michael|title=Calculus|publisher=Addison-Wesley|year=1967|chapter=15|pages=256–257|lccn=67-20770}}</ref> Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.

When [[radian]]s (rad) are employed, the angle is given as the length of the [[arc (geometry)|arc]] of the [[unit circle]] subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete [[turn (angle)|turn]] (360°) is an angle of 2{{pi}} (≈ 6.28) rad. For real number ''x'', the notationsnotation {{math|sin ''x''}}, {{math|cos ''x''}}, etc. referrefers to the value of the trigonometric functions evaluated at an angle of ''x'' rad. If units of degrees are intended, the degree sign must be explicitly shown (e.g., {{math|sin ''x°''}}, {{math|cos ''x°''}}, etc.). Using this standard notation, the argument ''x'' for the trigonometric functions satisfies the relationship ''x'' = (180''x''/{{pi}})°, so that, for example, {{math|1=sin {{pi}} = sin 180°}} when we take ''x'' = {{pi}}. In this way, the degree symbol can be regarded as a mathematical constant such that 1° = {{pi}}/180 ≈ 0.0175.

[[G. H. Hardy]] noted in his 1908 work ''[[A Course of Pure Mathematics]]'' that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number.<ref name="Hardy">{{citation|first=G.H.|last=Hardy|title=A 1908course pagesof 432-438pure mathematics|year=1950|edition=8th|pages=432–438}}</ref> Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.

HardyVarious describesways fourexist waysin ofthe literature for defining the trigonometric functions in a manner suitable for analysis; they include:

* Using the "geometry" of the unit circle, which requires formulating the arc length of a circle (or area of a sector) analytically.<ref name="Hardy"/>

* By a power series, which is particularly well-suited to complex variables.<ref name="Hardy"/><ref name="WW">Whittaker, E. T., & Watson, G. N. (1920). A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. University press.</ref>

* By using an infinite product expansion.<ref name="Hardy"/>

* By inverting the inverse trigonometric functions, which can be defined as integrals of algebraic or rational functions.<ref name="Hardy"/>

Another way to define the trigonometric functions in analysis is using integration.<ref> name="Hardy"/><ref>{{citation|last=Bartle|year=1964|title=Elements (1950),of pagereal 444analysis|publisher=|pages=315–316}}</ref> For a real number <math>t</math>, put

<math display="block">\sin\left(\theta + \frac{\pi}{2}\right) = \frac{-1}{t\sqrt{1+(-1/t)^2}} = \frac{-1}{\sqrt{1+t^2}} = -\cos(t\theta)</math>

<math display="block">\cos\left(\theta + \frac{\pi}{2}\right) = \frac{1}{\sqrt{1+(-1/t)^2}} = \frac{t}{\sqrt{1+t^2}} = \sin(t\theta).</math>

Many [[identity (mathematics)|identities]] interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see [[List of trigonometric identities]]. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval {{math|[0, {{pi}}/2]}}, see [[Proofs of trigonometric identities]]). For non-geometrical proofs using only tools of [[calculus]], one may use directly the differential equations, in a way that is similar to that of the [[#Relationship to exponential function (Euler's formula) and the exponential function|above proof]] of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.

<ref name="Klein_1924"klein>{{cite book |chapter=Die goniometrischen Funktionen |at={{nobr|Ch. 3.2}}, {{pgs|175 ff.}} |title=Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis |volume=1 |author-first=Christian Felix |author-last=Klein |author-link=Christian Felix Klein |date=1924<!-- 1927 --> |orig-year=1902<!-- 1908 --> |edition=3rd |publisher=[[J. Springer]] |location=Berlin |language=de |chapter-url=https://books.google.com/books?id=5t8fAAAAIAAJ&pg=PA175 }} Translated as {{cite book|title=Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis |author-first=Felix |author-last=Klein |author-link=Felix Klein |display-authors=0 |year=1932 |publisher=Macmillan |translator-first1=E. R. |translator-last1=Hedrick |translator-first2=C. A. |translator-last2=Noble |chapter-url=https://archive.org/details/geometryelementa0000feli/page/162/?q=%22ii.+the+goniometric+functions%22 |chapter=The Goniometric Functions |at=Ch. 3.2, {{pgs|162 ff.}} }}</ref>