What is tan − 1 ⁡ ( 5 i / 3 )My progress: Let tan...

Jovany Clayton

Jovany Clayton

Answered

2022-07-13

What is tan 1 ( 5 i / 3 )
My progress: Let tan x = 5 i 3 = sin x cos x I tried using sin x = e i x e i x 2 , cos x = e i x + e i x 2 to show that e i x e i x e i x + e i x = 5 3 or e 2 i x = 1 4 , but I'm stuck here.

Answer & Explanation

Kroatujon3

Kroatujon3

Expert

2022-07-14Added 19 answers

You are fine. Just solve e 2 i x = 1 4 , taking into account that x is complex. Write x=r+ic to get e 2 c + 2 i r = 1 4 = 1 4 e i π and identify r = π / 2 and c = ln 2. So
x = π / 2 + i ln 2

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