Introductions to Cos Introduction to the trigonometric functions General The six trigonometric functions sine sin z , cosine cos z , tangent tan z , cotangent cot z , cosecant csc z , and secant sec z are well known and among the most frequently used elementary functions. The most popular func- tions sin z , cos z , tan z , and cot z are taught worldwide in high school programs because of their natural appear- H L H L H L H L H L ance in problems involving angle measurement and their wide applications in the quantitative sciences. H L The trigonometricH L H L functionsH L shareH manyL common properties. Definitions of trigonometric functions All trigonometric functions can be defined as simple rational functions of the exponential function of ä z: ãä z - ã-ä z sin z 2 ä ãä z + ã-ä z cos z H L 2 ä ãä z - ã-ä z tanHzL - ãä z + ã-ä z ä ãIä z + ã-ä z M cotHzL ãä z - ã-ä z I 2 ä M cscHzL ãä z - ã-ä z 2 secHzL . ãä z + ã-ä z TheH Lfunctions tan z , cot z , csc z , and sec z can also be defined through the functions sin z and cos z using the following formulas: sin z H L H L H L H L H L H L tan z cos z cos z cot z H L H L sin z H L 1 csc z H L H L sin z H L 1 sec z . H L cos z H L A quickH L look at the trigonometric functions H L http://functions.wolfram.com 2 Here is a quick look at the graphics for the six trigonometric functions along the real axis. 3 2 1 sin x cos x f 0 -1 tan x H L -2 cot x H L -3 sec x -3 p -2 p -p 0 p 2 p 3 p cscHxL x H L H L Connections within the group of trigonometric functions andH Lwith other function groups Representations through more general functions The trigonometric functions are particular cases of more general functions. Among these more general functions, four different classes of special functions are particularly relevant: Bessel, Jacobi, Mathieu, and hypergeometric functions. For example, sin z and cos z have the following representations through Bessel, Mathieu, and hypergeometric functions: H L H L sin z p z J z sin z -ä p ä z I ä z sin z p z Y z sin z ä ä z K ä z - -ä z K -ä z 2 1 2 2 1 2 2 -1 2 1 2 1 2 2 p cos z p z J z cos z p ä z I ä z cos z - p z Y z cos z ä z K ä z + - ä z K -ä z H L 2 -1 2H L H L 2 -1 2 H L H L 2 1 2H L H L I2 p 1 2 H L 2 p 1 2 H LN sin z Se 1, 0, z cos z Ce 1, 0, z H L H L H L H L H L H L H L H L H L 3 z2 1 z2 sin z z 0F1 ; ; - cos z 0F1 ; ; - . H L H 2 L 4 H L H L 2 4 On the other hand, all trigonometric functions can be represented as degenerate cases of the corresponding doubly periodicH L JacobiJ ellipticN functionsH L whenJ theirN second parameter is equal to 0 or 1: sin z sd z 0 sn z 0 sin z -ä sc ä z 1 -ä sd ä z 1 cos z cd z 0 cn z 0 cos z nc ä z 1 nd ä z 1 tan z sc z 0 tan z -ä sn ä z 1 cotHzL csHz È 0L H È L cotHzL ä ns äHz 1È L H È L cscHzL dsHz È 0L ns Hz È0 L cscHzL ä csH ä zÈ 1L ä dsH äÈz L1 secHzL dcH zÈ 0L nc z 0 secHzL cn ä zH 1È L dn ä z 1 . H L H È L H L H È L RepresentationsH L H È L throughH È L relatedH L equivalentH È L functionsH È L H L H È L H È L H L H È L H È L Each of the six trigonometric functions can be represented through the corresponding hyperbolic function: sin z -ä sinh ä z sin ä z ä sinh z cos z cosh ä z cos ä z cosh z tan z -ä tanh ä z tan ä z ä tanh z cotHzL ä coth äHz L cotHä zL -ä cothH Lz cscHzL ä cschH ä zL cscHä zL -ä cschH L z secHzL sech ä zH L secHä zL sech Hz L. H L H L H L H L RelationsH L to HinverseL functionsH L H L H L H L H L H L http://functions.wolfram.com 3 Each of the six trigonometric functions is connected with its corresponding inverse trigonometric function by two formulas. One is a simple formula, and the other is much more complicated because of the multivalued nature of the inverse function: sin sin-1 z z sin-1 sin z z ; - p < Re z < p Re z - p Im z ³ 0 Re z p Im z £ 0 2 2 2 2 cos cos-1 z z cos-1 cos z z ; 0 < Re z < p Re z 0 Im z ³ 0 Re z p Im z £ 0 tan tan-1 z z tan-1 tan z z ; Re z < p Re z - p Im z < 0 Re z p Im z > 0 I H LM H H LL H2L ë H L 2 í H L ë H 2L í H L cotIcot-1 Hz LM z cot-1 Hcot Hz LL z ; Re z <H Lp ÞRe zH L - p ß ImH Lz < Þ0 HReL z ß p H LIm z ³ 0 2 2 2 cscIcsc-1HzLM z csc-1HcscHzLL z ; ReHzL¤ < p ë ReHzL - p í ImHzL £ 0ë ReHzL p í ImHzL ³ 0 2 2 2 secIsec-1HzLM z sec-1HsecHzLL z ; 0 <HReL¤ z <ëp ReH Lz 0 íIm z H³L 0 Reë z H Lp Imíz £ 0.H L I H LM H H LL H L¤ ë H L í H L ë H L í H L Representations through other trigonometric functions I H LM H H LL H L Þ H L ß H L Þ H L ß H L Each of the six trigonometric functions can be represented by any other trigonometric function as a rational func- tion of that function with linear arguments. For example, the sine function can be representative as a group-defining function because the other five functions can be expressed as follows: cos z sin p - z cos2 z 1 - sin2 z 2 2 sin z sin z 2 sin z tan z p tan z cos z sin -z 1-sin2 z 2 H L I M p H L H L H L sin H-Lz H2 L cos z 2 2 1-sin z cot z H L J N cot z H L H L sin z sin z H L sin2 z 1 1 csc z H L J N csc2 z H L sin z 2 H L H L sin z H L H L H L sec z 1 1 sec2 z 1 . cos z p 2 H L sin -z 1-sin z H L 2 H L H L All six trigonometricH L functions can beH L transformed into any other trigonometric function of this group if the H L J N H L argument z is replaced by p p 2 + q z with q2 1 p Î Z: sin -z - 2 p -sin z sin z - 2 p sin z sin -z - 3 p cos z sin z - 3 p cos z ì 2 2 sinH-z - p L sin z H L sinHz - p L -sinH zL sin -z - p -cos z sin z - p -cos z J 2 N H L J 2 N H L sin z + p cos z sin p - z cos z H 2 L H L H 2 L H L sinIz + p M -sin z H L sinIp - z M sin z H L sin z + 3 p -cos z sin 3 p - z -cos z I 2M H L I 2 M H L sinHz + 2Lp sin Hz L sinH2 p -Lz -HsinL z J N H L J N H L H L H L H L H L http://functions.wolfram.com 4 cos -z - 2 p cos z cos z - 2 p cos z cos -z - 3 p sin z cos z - 3 p -sin z 2 2 cosH-z - p L -cosHzL cos Hz - p L -cos HzL cos -z - p -sin z cos z - p sin z J 2 N H L J 2 N H L cos z + p -sin z cos p - z sin z H 2 L H L H 2 L H L cosIz + p M -cos zH L cosIp - z M -cosH L z cos z + 3 p sin z cos 3 p - z -sin z I 2M H L I 2 M H L cosHz + 2Lp cos Hz L cosH2 p -Lz cos Hz L tanJ-z - pN -tanH L z tan Jz - p N tan z H L tanH-z - p L cotHzL tan zH - p L -cot Hz L 2 2 tan z + p -cot z tan p - z cot z H 2 L H L H2 L H L tanIz + p M tan zH L tanI p - zM -tanHzL cotI-z - Mp -cotH Lz cotI z - pM cotHzL cotH-z -Lp tanH Lz cot Hz - p L -tan H zL 2 2 cot z + p -tan z cot p - z tan z H 2 L H L H2 L H L cotIz + p M cot zH L cotIp - z M -cot Hz L cscI-z -M2 p -HcscL z cscI z -M2 p HcscL z cscH-z -L3 p HsecL z cscH z -L 3 p secH L z 2 2 cscH-z - p L csc z H L csc Hz - p L -csc HzL csc -z - p -sec z csc z - p -sec z J 2 N H L J 2 N H L csc z + p sec z csc p - z sec z H 2 L H L H 2 L H L cscIz + p M -csc zH L cscIp - z M csc zH L csc z + 3 p -sec z csc 3 p - z -sec z I 2M H L I 2 M H L cscHz + 2Lp csc Hz L csc H2 p -Lz -cscH L z secJ-z - 2Np secHzL sec zJ - 2 p N sec z H L secH-z - 3Lp cscH Lz sec zH- 3 p L -csc zH L 2 2 secH-z - p L -secHzL sec Hz - p L -sec HzL sec -z - p -csc z sec z - p csc z J 2 N H L J 2 N H L sec z + p -csc z sec p - z csc z H 2 L H L H 2 L H L secIz + p M -sec zH L secIp - z M -secH Lz sec z + 3 p csc z sec 3 p - z -csc z I 2M H L I 2 M H L secHz + 2Lp sec Hz L sec H2 p -Lz sec Hz L.