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Math Help - Solve cos z = 5

  1. #1
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    Solve cos z = 5

    HOW DO I DO THIS?!

    Solve cos(z) = 5

    Help... please ...

    need someone to go through slowly with me lol :S
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  2. #2
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    Re: Solve cos z = 5

    You should note that \displaystyle \cos{(z)} = \cos{(x + iy)} = \cos{(x)}\cosh{(y)} - i\sin{(x)}\sinh{(y)}, so

    \displaystyle \cos{(x)}\cosh{(y)} = 5 and \displaystyle -\sin{(x)}\sinh{(y)} = 0.

    From the second equation, we can see that either \displaystyle \sin{(x)} = 0 \implies x = n\pi, n \in \mathbf{Z} or \displaystyle \sinh{(y)} = 0 \implies y = 0.

    If \displaystyle x = n\pi then

    \displaystyle \begin{align*} \cos{(x)}\cosh{(y)} &= 5 \\ \cos{(n\pi)}\cosh{(y)} &= 5 \\ (-1)^n\cosh{(y)} &= 5 \\ \cosh{(y)} &= \frac{5}{(-1)^n} \end{align*}

    and since the hyperbolic cosine is always positive, that means we can only accept even values for \displaystyle n, so \displaystyle n = 2m, m\in \mathbf{Z} and \displaystyle y = \textrm{arcosh}\,{(5)}.

    So the first solution is \displaystyle z = 2m\pi + \textrm{arcosh}\,{(5)}, m \in \mathbf{Z}.

    See if you can get the second solution, for where \displaystyle y = 0.
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