Find all z such that | tan ⁡ z | = 1

opepayflarpws

opepayflarpws

Answered

2022-06-24

Find all z such that | tan z | = 1

Answer & Explanation

Jaylee Dodson

Jaylee Dodson

Expert

2022-06-25Added 22 answers

tan z = sin z cos z = e i z e i z i ( e i z + e i z )
So
| tan z | = | e i z e i z e i z + e i z | = 1
| e i z + e i z | = | e i z e i z |
| e 2 i z + 1 | 2 = | e 2 i z 1 | 2
( e 2 i z + 1 ) ( e 2 i z + 1 ) ¯ = ( e 2 i z 1 ) ( e 2 i z 1 ) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
( e 2 i z + 1 ) ( e 2 i z + 1 ) = ( e 2 i z 1 ) ( e 2 i z 1 )
1 + e 2 i z + e 2 i z + 1 = 1 e 2 i z e 2 i z + 1
e 2 i z + e 2 i z = 0
cos ( 2 z ) + i sin ( 2 z ) + cos ( 2 z ) i sin ( 2 z ) = 0
cos ( 2 z ) = 0
So
2 z = n π + π 2

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