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`Complex FunctionβComplex Trigonometric FunctionβProperties of Sine and Cosine`

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# Complex Function

## Complex Trigonometric Function

For the function, β―ππ=cosπ+πsinπ

```Let β―ππ=cosπ+πsinπ Therefore, β―βππ=cos(βπ)+πsin(βπ)=cosπβπsinπ Hence, β―ππ+β―βππ=2cosπ and β―ππββ―βππ=2πsinπ Thus cosπ=β―ππ+β―βππ2 and sinπ=β―ππββ―βππ2π```

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

`cosπ§=β―ππ§+β―βππ§2 and sinπ§=β―ππ§ββ―βππ§2π`

## Properties of Sine and Cosine

For the function, cosπ§=β―ππ§+β―βππ§2 and sinπ§=β―ππ§ββ―βππ§2π

• cosπ§ and sinπ§ are analytic functions (in fact, entire).
• For real-valued π§, (i.e. π§=π₯+πβ0) the complex sine and cosine agree with the real-valued sine and cosine functions.
• cos(βπ§)=β―βππ§+β―ππ§2=cosπ§
• sin(βπ§)=β―βππ§ββ―ππ§2π=βsin𝑧
• cos(π§+π€)=cosπ§cosπ€βsinπ§sinπ€, sin(π§+π€)=sinπ§cosπ€+cosπ§sinπ€

Proofs of the addition formulae cos(π§+π€)

``` cosπ§cosπ€βsinπ§sinπ€ =(β―ππ§+β―βππ§2)(β―ππ€+β―βππ€2) β(β―ππ§ββ―βππ§2π)(β―ππ€ββ―βππ€2π) =(β―ππ§+β―βππ§)(β―ππ€+β―βππ€)+(β―ππ§ββ―βππ§)(β―ππ€ββ―βππ€)4 =β―ππ§β―ππ€+β―ππ§β―βππ€+β―βππ§β―ππ€+β―βππ§β―βππ€+β―ππ§β―ππ€ββ―ππ§β―βππ€ββ―βππ§β―ππ€+β―βππ§β―βππ€4 =2β―ππ§β―ππ€+2β―βππ§β―βππ€4 =β―π(π§+π€)+β―βπ(π§+π€)2 =cos(π§+π€)```
• cos(π§+2π)=β―π(π§+2π)+β―βπ(π§+2π)2=cosπ§
• sin(π§+2π)=β―π(π§+2π)ββ―βπ(π§+2π)2π=sinπ§
• cos2π§+cos2π§=1. Proof: Let π€=βπ§ in the addition formula for cosine.
• sin(π§+π2)=cosπ§

Proof:

```sin(π§+π2)=β―π(π§+π2)ββ―βπ(π§+π2)2π  =πβ―ππ§β(βπ)β―βππ§2π  =β―ππ§+β―βππ§2=cosπ§ ```
• sinπ§=0βπ§=ππ, πββ€

Proof

```sinπ§=0ββ―ππ§ββ―βππ§2π=0  ββ―ππ§=β―βππ§  βππ§β(βππ§)=2πππ, πββ€, the periodicity of the exponential with period of 2πππ.  β2ππ§=2πππ, πββ€  βπ§=ππ, πββ€ ```
• cosπ§=0βπ§=π2+ππ, πββ€

Proof

```cosπ§=0ββ―ππ§+β―βππ§2=0  ββ―2ππ§+12=(β―ππ§+π)(β―ππ§βπ)2=0  ββ―2ππ§+1=0  ββ―2ππ§=β1=β―ππ  β2ππ§βππ=2πππ, πββ€, the periodicity of the exponential with period of 2πππ  β2ππ§=(2π+1)ππ, πββ€  βπ§=π2+ππ, πββ€ ```
• Derivative of Sine: πππ§sinπ§=cosπ§

Proof

```πππ§sinπ§=πππ§β―ππ§ββ―βππ§2π  =πβ―ππ§β(βπ)β―βππ§2π  =β―ππ§+β―βππ§2=cosπ§ ```
• Derivative of Cosine: `πππ§cosπ§=βsinπ§`

Proof

```πππ§cosπ§=πππ§β―ππ§+β―βππ§2  =πβ―ππ§+(βπ)β―βππ§2  =π(β―ππ§ββ―βππ§)2=βsinπ§ ```
• Complex sine in terms of real functions, `sinπ§=sinπ₯coshπ¦+πcosπ₯sinhπ¦`

Proof

```sinπ§=sin(π₯+ππ¦)  =sinπ₯cos(ππ¦)+cosπ₯sin(ππ¦)  =sinπ₯β―π(ππ¦)+β―βπ(ππ¦)2+cosπ₯β―π(ππ¦)ββ―βπ(ππ¦)2π  =sinπ₯β―βπ¦+β―π¦2+cosπ₯β―βπ¦ββ―π¦2π  =sinπ₯β―π¦+β―βπ¦2+πcosπ₯β―π¦ββ―βπ¦2  =sinπ₯coshπ¦+πcosπ₯sinhπ¦ ```
• Complex cosine in terms of real functions, `cosπ§=cosπ₯coshπ¦+πsinπ₯sinhπ¦`

Proof

```cosπ§=cos(π₯+ππ¦)  =cosπ₯cos(ππ¦)βsinπ₯sin(ππ¦)  =cosπ₯β―π(ππ¦)+β―βπ(ππ¦)2+sinπ₯β―π(ππ¦)ββ―βπ(ππ¦)2π  =cosπ₯β―βπ¦+β―π¦2+sinπ₯β―βπ¦ββ―π¦2π  =cosπ₯β―π¦+β―βπ¦2+πsinπ₯β―π¦ββ―βπ¦2  =cosπ₯coshπ¦+πsinπ₯sinhπ¦ ```

ID: 190400011 Last Updated: 11/4/2019 Revision: 0

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