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1: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
§4.23(i) General Definitions
§4.23(ii) Principal Values
§4.23(iv) Logarithmic Forms
§4.23(vii) Special Values and Interrelations
2: 4.27 Sums
§4.27 Sums
For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).
3: 4.47 Approximations
§4.47 Approximations
Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for ln , exp , sin , cos , tan , cot , arcsin , arctan , arcsinh . Schonfelder (1980) gives 40D coefficients for sin , cos , tan . … Hart et al. (1968) give ln , exp , sin , cos , tan , cot , arcsin , arccos , arctan , sinh , cosh , tanh , arcsinh , arccosh . … Luke (1975, Chapter 3) supplies real and complex approximations for ln , exp , sin , cos , tan , arctan , arcsinh . …
4: 4.1 Special Notation
The main purpose of the present chapter is to extend these definitions and properties to complex arguments z . The main functions treated in this chapter are the logarithm ln z , Ln z ; the exponential exp z , e z ; the circular trigonometric (or just trigonometric) functions sin z , cos z , tan z , csc z , sec z , cot z ; the inverse trigonometric functions arcsin z , Arcsin z , etc. ; the hyperbolic trigonometric (or just hyperbolic) functions sinh z , cosh z , tanh z , csch z , sech z , coth z ; the inverse hyperbolic functions arcsinh z , Arcsinh z , etc. Sometimes in the literature the meanings of ln and Ln are interchanged; similarly for arcsin z and Arcsin z , etc. … sin 1 z for arcsin z and Sin 1 z for Arcsin z .
5: 4.35 Identities
4.35.14 2 sinh u sinh v = cosh ( u + v ) cosh ( u v ) ,
4.35.16 2 sinh u cosh v = sinh ( u + v ) + sinh ( u v ) .
4.35.18 cosh 2 u cosh 2 v = sinh ( u + v ) sinh ( u v ) ,
4.35.34 sinh z = sinh x cos y + i cosh x sin y ,
4.35.35 cosh z = cosh x cos y + i sinh x sin y ,
6: 4.14 Definitions and Periodicity
§4.14 Definitions and Periodicity
4.14.7 cot z = cos z sin z = 1 tan z .
The functions sin z and cos z are entire. In the zeros of sin z are z = k π , k ; the zeros of cos z are z = ( k + 1 2 ) π , k . The functions tan z , csc z , sec z , and cot z are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). …
7: 4.21 Identities
§4.21 Identities
§4.21(i) Addition Formulas
§4.21(ii) Squares and Products
§4.21(iii) Multiples of the Argument
§4.21(iv) Real and Imaginary Parts; Moduli
8: 4.28 Definitions and Periodicity
Relations to Trigonometric Functions
As a consequence, many properties of the hyperbolic functions follow immediately from the corresponding properties of the trigonometric functions.
Periodicity and Zeros
The functions sinh z and cosh z have period 2 π i , and tanh z has period π i . The zeros of sinh z and cosh z are z = i k π and z = i ( k + 1 2 ) π , respectively, k .
9: 4.16 Elementary Properties
§4.16 Elementary Properties
Table 4.16.1: Signs of the trigonometric functions in the four quadrants.
Quadrant sin θ , csc θ cos θ , sec θ tan θ , cot θ
Table 4.16.2: Trigonometric functions: quarter periods and change of sign.
x θ 1 2 π ± θ π ± θ 3 2 π ± θ 2 π ± θ
Table 4.16.3: Trigonometric functions: interrelations. …
sin θ = a cos θ = a tan θ = a csc θ = a sec θ = a cot θ = a
10: 4.32 Inequalities
4.32.1 cosh x ( sinh x x ) 3 ,
4.32.2 sin x cos x < tanh x < x , x > 0 ,
4.32.3 | cosh x cosh y | | x y | sinh x sinh y , x > 0 , y > 0 ,
4.32.4 arctan x 1 2 π tanh x , x 0 .