Editing Trigonometric functions (section)

==Radians versus degrees==
In geometric applications, the argument of a trigonometric function is generally the measure of an [[angle]]. For this purpose, any [[angular unit]] is convenient. One common unit is [[degree (angle)|degrees]], in which a right angle is 90° and a complete turn is 360° (particularly in [[elementary mathematics]]).

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 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°),<ref>{{cite oeis|A072097|Decimal expansion of 180/Pi}}</ref> and a complete [[turn (angle)|turn]] (360°) is an angle of 2{{pi}} (≈ 6.28) rad.<ref>{{cite oeis|A019692|Decimal expansion of 2*Pi}}</ref> For real number ''x'', the notation {{math|sin ''x''}}, {{math|cos ''x''}}, etc. refers 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 ({{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.<ref>{{cite oeis|A019685|Decimal expansion of Pi/180}}</ref>