http://google.com, pub-7771400403364887, DIRECT, f08c47fec0942fa0 find all values of z such that eˣ=-2

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find all values of z such that eˣ=-2

Solution 
Let z be =x 

eˣ=-2 implies that  |-2|=2 and arg (-2)= 180°eˣ=2(cos180°+is in180°)

eⁱˣ=cosx+isinx, eⁱˣ=cosx-isinx 

adding the two gives 

eⁱˣ+eⁱˣ=2cosx⇒cosx=(eⁱˣ+eⁱˣ)/2

Subtracting gives 

.
eⁱˣ-eⁱˣ=2isinx⇒sinx=(eⁱˣ-e⁻ⁱˣ)/2i

Cosx =(eⁱˣ+eˣ)/2

Cosx=(eⁱˣ+e⁻ˣ)/2

sinx=(eⁱˣ-e⁻ⁱˣ)/2i

For every real x it is therefore nature to define the sine and cosine functions of a complex variable z by the equations 

Sinz =(eⁱˣ-e⁻ⁱˣ)/2i

Cos =(eⁱˣ+e⁻ⁱˣ)/2

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