4 Elementary Functions Trigonometric Functions 4.13 Lambert $\mathop{W\/}\nolimits$ -Function 4.15 Graphics

§4.14 Definitions and Periodicity

Notes:: See Levinson and Redheffer (1970, pp. 55–57).
Keywords:: trigonometric functions
Referenced by:: ¶ ‣ §1.9(i)
Permalink:: http://dlmf.nist.gov/4.14

4.14.1 $\mathop{\sin\/}\nolimits z=\frac{e^{{iz}}-e^{{-iz}}}{2i},$

Defines:: $\mathop{\sin\/}\nolimits z$ : sine function
Symbols:: : base of exponential function and : complex variable
A&S Ref:: 4.3.1
Referenced by:: §4.45(ii)
Permalink:: http://dlmf.nist.gov/4.14.E1
Encodings:: TeX, pMML, png

4.14.2 $\mathop{\cos\/}\nolimits z=\frac{e^{{iz}}+e^{{-iz}}}{2},$

Defines:: $\mathop{\cos\/}\nolimits z$ : cosine function
Symbols:: : base of exponential function and : complex variable
A&S Ref:: 4.3.2
Permalink:: http://dlmf.nist.gov/4.14.E2
Encodings:: TeX, pMML, png

4.14.3 $\mathop{\cos\/}\nolimits z\pm i\mathop{\sin\/}\nolimits z=e^{{\pm iz}},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function and : complex variable
Permalink:: http://dlmf.nist.gov/4.14.E3
Encodings:: TeX, pMML, png

4.14.4 $\mathop{\tan\/}\nolimits z=\frac{\mathop{\sin\/}\nolimits z}{\mathop{\cos\/}% \nolimits z},$

Defines:: $\mathop{\tan\/}\nolimits z$ : tangent function
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and : complex variable
A&S Ref:: 4.3.3
Referenced by:: §4.14, ¶ ‣ §4.45(i)
Permalink:: http://dlmf.nist.gov/4.14.E4
Encodings:: TeX, pMML, png

4.14.5 $\mathop{\csc\/}\nolimits z=\frac{1}{\mathop{\sin\/}\nolimits z},$

Defines:: $\mathop{\csc\/}\nolimits z$ : cosecant function
Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function and : complex variable
A&S Ref:: 4.3.4
Permalink:: http://dlmf.nist.gov/4.14.E5
Encodings:: TeX, pMML, png

4.14.6 $\mathop{\sec\/}\nolimits z=\frac{1}{\mathop{\cos\/}\nolimits z},$

Defines:: $\mathop{\sec\/}\nolimits z$ : secant function
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and : complex variable
A&S Ref:: 4.3.5
Permalink:: http://dlmf.nist.gov/4.14.E6
Encodings:: TeX, pMML, png

4.14.7 $\mathop{\cot\/}\nolimits z=\frac{\mathop{\cos\/}\nolimits z}{\mathop{\sin\/}% \nolimits z}=\frac{1}{\mathop{\tan\/}\nolimits z}.$

Defines:: $\mathop{\cot\/}\nolimits z$ : cotangent function
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : tangent function and : complex variable
A&S Ref:: 4.3.6
Referenced by:: §4.14, ¶ ‣ §4.45(i), §4.45(ii)
Permalink:: http://dlmf.nist.gov/4.14.E7
Encodings:: TeX, pMML, png

The functions $\mathop{\sin\/}\nolimits z$ and $\mathop{\cos\/}\nolimits z$ are entire. In $\Complex$ the zeros of $\mathop{\sin\/}\nolimits z$ are $z=k\pi$ , $k\in\Integer$ ; the zeros of $\mathop{\cos\/}\nolimits z$ are $z=\left(k+\tfrac{1}{2}\right)\pi$ , $k\in\Integer$ . The functions $\mathop{\tan\/}\nolimits z$ , $\mathop{\csc\/}\nolimits z$ , $\mathop{\sec\/}\nolimits z$ , and $\mathop{\cot\/}\nolimits z$ are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7).

For $k\in\Integer$

4.14.8 $\mathop{\sin\/}\nolimits\!\left(z+2k\pi\right)=\mathop{\sin\/}\nolimits z,$

Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function, : integer and : complex variable
A&S Ref:: 4.3.7
Permalink:: http://dlmf.nist.gov/4.14.E8
Encodings:: TeX, pMML, png

4.14.9 $\mathop{\cos\/}\nolimits\!\left(z+2k\pi\right)=\mathop{\cos\/}\nolimits z,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : integer and : complex variable
A&S Ref:: 4.3.8
Permalink:: http://dlmf.nist.gov/4.14.E9
Encodings:: TeX, pMML, png

4.14.10 $\mathop{\tan\/}\nolimits\!\left(z+k\pi\right)=\mathop{\tan\/}\nolimits z.$

Symbols:: $\mathop{\tan\/}\nolimits z$ : tangent function, : integer and : complex variable
A&S Ref:: 4.3.9
Permalink:: http://dlmf.nist.gov/4.14.E10
Encodings:: TeX, pMML, png

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