tan &#x2061;<!-- ⁡ --> ( x + y i ) = &#x03B1;<!-- α --> + &#x03

DIAMMIBENVERMk1

DIAMMIBENVERMk1

Answered question

2022-07-12

tan ( x + y i ) = α + β i tan ( x y i ) = α β i?

Answer & Explanation

Danika Rojas

Danika Rojas

Beginner2022-07-13Added 9 answers

I'd use that
sin ( x ) = 1 2 i ( e i x e i x ) and cos ( x ) = 1 2 ( e i x + e i x ) .
Then,
tan ( x + i y ) = e i x y e i x + y i ( e i x y + e i x + y ) = e y e i x e y e i x i ( e y e i x + e y e i x )
Taking conjugates
tan ( x + i y ) ¯ = e y e i x e y e i x i ( e y e i x + e y e i x ) = e y e i x e y e i x i ( e y e i x + e y e i x ) .
On the other hand,
tan ( x i y ) = e i x + y e i x y i ( e i x + y + e i x y ) = e y e i x e y e i x i ( e y e i x + e y e i x ) .
And we observe that tan ( x + i y ) ¯ = tan ( x i y ), the original statement.

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