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Complex Function
 Complex Trigonometric Function
 Properties of Sine and Cosine

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Complex Function

Complex Trigonometric Function

For the function, β„―π‘–πœƒ=cosπœƒ+𝑖sinπœƒ

Let β„―π‘–πœƒ=cosπœƒ+𝑖sinπœƒ Therefore, β„―βˆ’π‘–πœƒ=cos(βˆ’πœƒ)+𝑖sin(βˆ’πœƒ)=cosπœƒβˆ’π‘–sinπœƒ Hence, β„―π‘–πœƒ+β„―βˆ’π‘–πœƒ=2cosπœƒ and β„―π‘–πœƒβˆ’β„―βˆ’π‘–πœƒ=2𝑖sinπœƒ Thus cosπœƒ=β„―π‘–πœƒ+β„―βˆ’π‘–πœƒ2 and sinπœƒ=β„―π‘–πœƒβˆ’β„―βˆ’π‘–πœƒ2𝑖

By definition, The complex cosine and sine functions are defined via

cos𝑧=ℯ𝑖𝑧+β„―βˆ’π‘–π‘§2 and sin𝑧=β„―π‘–π‘§βˆ’β„―βˆ’π‘–π‘§2𝑖

Properties of Sine and Cosine

For the function, cos𝑧=ℯ𝑖𝑧+β„―βˆ’π‘–π‘§2 and sin𝑧=β„―π‘–π‘§βˆ’β„―βˆ’π‘–π‘§2𝑖

  • cos𝑧 and sin𝑧 are analytic functions (in fact, entire).
  • For real-valued 𝑧, (i.e. 𝑧=π‘₯+𝑖⋅0) the complex sine and cosine agree with the real-valued sine and cosine functions.
  • cos(βˆ’π‘§)=β„―βˆ’π‘–π‘§+ℯ𝑖𝑧2=cos𝑧
  • sin(βˆ’π‘§)=β„―βˆ’π‘–π‘§βˆ’β„―π‘–π‘§2𝑖=βˆ’sin𝑧
  • cos(𝑧+𝑀)=cos𝑧cosπ‘€βˆ’sin𝑧sin𝑀, sin(𝑧+𝑀)=sin𝑧cos𝑀+cos𝑧sin𝑀

    Proofs of the addition formulae cos(𝑧+𝑀)

     cos𝑧cosπ‘€βˆ’sin𝑧sin𝑀 =(ℯ𝑖𝑧+β„―βˆ’π‘–π‘§2)(ℯ𝑖𝑀+β„―βˆ’π‘–π‘€2) βˆ’(β„―π‘–π‘§βˆ’β„―βˆ’π‘–π‘§2𝑖)(β„―π‘–π‘€βˆ’β„―βˆ’π‘–π‘€2𝑖) =(ℯ𝑖𝑧+β„―βˆ’π‘–π‘§)(ℯ𝑖𝑀+β„―βˆ’π‘–π‘€)+(β„―π‘–π‘§βˆ’β„―βˆ’π‘–π‘§)(β„―π‘–π‘€βˆ’β„―βˆ’π‘–π‘€)4 =ℯ𝑖𝑧ℯ𝑖𝑀+β„―π‘–π‘§β„―βˆ’π‘–π‘€+β„―βˆ’π‘–π‘§β„―π‘–π‘€+β„―βˆ’π‘–π‘§β„―βˆ’π‘–π‘€+β„―π‘–π‘§β„―π‘–π‘€βˆ’β„―π‘–π‘§β„―βˆ’π‘–π‘€βˆ’β„―βˆ’π‘–π‘§β„―π‘–π‘€+β„―βˆ’π‘–π‘§β„―βˆ’π‘–π‘€4 =2ℯ𝑖𝑧ℯ𝑖𝑀+2β„―βˆ’π‘–π‘§β„―βˆ’π‘–π‘€4 =ℯ𝑖(𝑧+𝑀)+β„―βˆ’π‘–(𝑧+𝑀)2 =cos(𝑧+𝑀)
  • cos(𝑧+2πœ‹)=ℯ𝑖(𝑧+2πœ‹)+β„―βˆ’π‘–(𝑧+2πœ‹)2=cos𝑧
  • sin(𝑧+2πœ‹)=ℯ𝑖(𝑧+2πœ‹)βˆ’β„―βˆ’π‘–(𝑧+2πœ‹)2𝑖=sin𝑧
  • cos2𝑧+cos2𝑧=1. Proof: Let 𝑀=βˆ’π‘§ in the addition formula for cosine.
  • sin(𝑧+πœ‹2)=cos𝑧

    Proof:

    sin(𝑧+πœ‹2)=ℯ𝑖(𝑧+πœ‹2)βˆ’β„―βˆ’π‘–(𝑧+πœ‹2)2𝑖  =π‘–β„―π‘–π‘§βˆ’(βˆ’π‘–)β„―βˆ’π‘–π‘§2𝑖  =ℯ𝑖𝑧+β„―βˆ’π‘–π‘§2=cos𝑧
  • sin𝑧=0⇔𝑧=π‘˜πœ‹, π‘˜βˆˆβ„€

    Proof

    sin𝑧=0β‡”β„―π‘–π‘§βˆ’β„―βˆ’π‘–π‘§2𝑖=0  β‡”β„―𝑖𝑧=β„―βˆ’π‘–π‘§  β‡”π‘–π‘§βˆ’(βˆ’π‘–π‘§)=2π‘˜πœ‹π‘–, π‘˜βˆˆβ„€, the periodicity of the exponential with period of 2π‘˜πœ‹π‘–.  β‡”2𝑖𝑧=2π‘˜πœ‹π‘–, π‘˜βˆˆβ„€  β‡”𝑧=π‘˜πœ‹, π‘˜βˆˆβ„€
  • cos𝑧=0⇔𝑧=πœ‹2+π‘˜πœ‹, π‘˜βˆˆβ„€

    Proof

    cos𝑧=0⇔ℯ𝑖𝑧+β„―βˆ’π‘–π‘§2=0  β‡”β„―2𝑖𝑧+12=(ℯ𝑖𝑧+𝑖)(β„―π‘–π‘§βˆ’π‘–)2=0  β‡”β„―2𝑖𝑧+1=0  β‡”β„―2𝑖𝑧=βˆ’1=β„―πœ‹π‘–  β‡”2π‘–π‘§βˆ’πœ‹π‘–=2π‘˜πœ‹π‘–, π‘˜βˆˆβ„€, the periodicity of the exponential with period of 2π‘˜πœ‹π‘–  β‡”2𝑖𝑧=(2π‘˜+1)πœ‹π‘–, π‘˜βˆˆβ„€  β‡”𝑧=πœ‹2+π‘˜πœ‹, π‘˜βˆˆβ„€
  • Derivative of Sine: 𝑑𝑑𝑧sin𝑧=cos𝑧

    Proof

    𝑑𝑑𝑧sin𝑧=π‘‘π‘‘π‘§β„―π‘–π‘§βˆ’β„―βˆ’π‘–π‘§2𝑖  =π‘–β„―π‘–π‘§βˆ’(βˆ’π‘–)β„―βˆ’π‘–π‘§2𝑖  =ℯ𝑖𝑧+β„―βˆ’π‘–π‘§2=cos𝑧
  • Derivative of Cosine: 𝑑𝑑𝑧cos𝑧=βˆ’sin𝑧

    Proof

    𝑑𝑑𝑧cos𝑧=𝑑𝑑𝑧ℯ𝑖𝑧+β„―βˆ’π‘–π‘§2  =𝑖ℯ𝑖𝑧+(βˆ’π‘–)β„―βˆ’π‘–π‘§2  =𝑖(β„―π‘–π‘§βˆ’β„―βˆ’π‘–π‘§)2=βˆ’sin𝑧
  • Complex sine in terms of real functions, sin𝑧=sinπ‘₯cosh𝑦+𝑖cosπ‘₯sinh𝑦

    Proof

    sin𝑧=sin(π‘₯+𝑖𝑦)  =sinπ‘₯cos(𝑖𝑦)+cosπ‘₯sin(𝑖𝑦)  =sinπ‘₯ℯ𝑖(𝑖𝑦)+β„―βˆ’π‘–(𝑖𝑦)2+cosπ‘₯ℯ𝑖(𝑖𝑦)βˆ’β„―βˆ’π‘–(𝑖𝑦)2𝑖  =sinπ‘₯β„―βˆ’π‘¦+ℯ𝑦2+cosπ‘₯β„―βˆ’π‘¦βˆ’β„―π‘¦2𝑖  =sinπ‘₯ℯ𝑦+β„―βˆ’π‘¦2+𝑖cosπ‘₯β„―π‘¦βˆ’β„―βˆ’π‘¦2  =sinπ‘₯cosh𝑦+𝑖cosπ‘₯sinh𝑦
  • Complex cosine in terms of real functions, cos𝑧=cosπ‘₯cosh𝑦+𝑖sinπ‘₯sinh𝑦

    Proof

    cos𝑧=cos(π‘₯+𝑖𝑦)  =cosπ‘₯cos(𝑖𝑦)βˆ’sinπ‘₯sin(𝑖𝑦)  =cosπ‘₯ℯ𝑖(𝑖𝑦)+β„―βˆ’π‘–(𝑖𝑦)2+sinπ‘₯ℯ𝑖(𝑖𝑦)βˆ’β„―βˆ’π‘–(𝑖𝑦)2𝑖  =cosπ‘₯β„―βˆ’π‘¦+ℯ𝑦2+sinπ‘₯β„―βˆ’π‘¦βˆ’β„―π‘¦2𝑖  =cosπ‘₯ℯ𝑦+β„―βˆ’π‘¦2+𝑖sinπ‘₯β„―π‘¦βˆ’β„―βˆ’π‘¦2  =cosπ‘₯cosh𝑦+𝑖sinπ‘₯sinh𝑦

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ID: 190400011 Last Updated: 2019/4/11 Revision:

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