# Thread: Solve cos z = 5

1. ## Solve cos z = 5

HOW DO I DO THIS?!

Solve $cos(z) = 5$

need someone to go through slowly with me lol :S

2. ## 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 $\displaystyle y = 0$.

,

,

,

,

### solve cos(z)=5

Click on a term to search for related topics.