Key to "What are the solutions for the equations cos⁡Z=−i,3sin⁡Z+icos⁡Z=eiz,and"

tripiverded9

Answered question

2022-01-17

What are the solutions for the equations

\cos Z = - i, 3 \sin Z + i \cos Z = e^{i z}, and e^{2 z} + 4 i = - 3 \sqrt{3} - 3 i

Bob Huerta

Beginner2022-01-18Added 41 answers

\cos z = i

\cos^{2} z = - 1

\sin^{2} z = 2

\sin z = \pm \sqrt{2}

\cos z + i \sin z = - 1 \pm i \sqrt{2}

e^{i z} = \sqrt{3} e^{\pm i (π - \arctan \sqrt{2}) + 2 n π i}

i z = \ln \sqrt{3} \pm i (π - \arctan \sqrt{2} + 2 n π)

z = \pm \arctan \sqrt{2} + (2 n + 1) π - i \ln \sqrt{3}, n \in Z

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You should note that there are other ways of solving the above equation, but however you solve it, the solution sets should be the same.

Orlando Paz

Beginner2022-01-19Added 42 answers

The last one is a linear equation in

e^{2 z}

, which can be solved by taking logarithms. For the others, write everything in terms of

e^{i z} and e^{- i z} = \frac{1}{e^{i z}}

, giving a quadratic equation.

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