# Complex trig functions and images

Does $\sin(z)$ omit any values in $\mathbb{C}$?

# My thoughts

Obviously it takes on all real value between $-1$ and $1$. Furthermore, I can show that it takes on another real value such as $4$.I believe $\sin(z)$ will not omit any values since I will always end up with a quadratic which is always solvable in $\mathbb{C}$. Is this correct?

Does $\tan(z)$ omit any values in $\mathbb{C}$?

My guess is similar to above.

Am I on the right track?

# Edit

Here is some of previous work as my justification

Find all values of $z$ such that $\tan(z)=3i$

\begin{align*} \frac{i(e^{iz}+e^{-iz})}{e^{iz}-e^{-iz}}&=3i\\ e^{iz}+e^{-iz}&=3(e^{iz}-e^{-iz})\\ -2e^{iz}&=-4e^{-iz}\\ e^{iz}&=2e^{-iz}\\ e^{-iz}(e^{2iz}-2)&=0\\ e^{2iz}&=2 \end{align*}

So we have $e^{-2y}=2$ so $y=-\frac{\ln(2)}{2}$ and $x=0+2\pi k$ since $\arg(2)=0$ and $\arg (e^{2iz})=2x$

Find all values of $\sin(z)=2$

We consider the exponential formulation of $\sin(z)$ and we have

\begin{align*} \frac{e^{iz}-e^{-iz}}{2i}&=3\\ e^{iz}-e^{-iz}&=6i\\ e^{-iz}(e^{2iz}-1)&=6i\\ e^{2iz}-1&=6ie^{iz} \tag{let w=e^{iz}}\\ w^2-6iw-1&=0 \end{align*}

From the quadratic equation we have

$$\frac{6i \pm \sqrt{-36+4}}{2}=3i \pm 2\sqrt{2}i$$

so $e^{iz}=(3 \pm 2\sqrt{2})i$

• Unfortunately, proving that some values are attained is not sufficient, but of course it can give ideas about how to prove it for all or for finding the ones that are missed. – egreg Feb 28 '16 at 22:44

You want to solve the equation $\sin z=w$. Write $$\sin z=\frac{e^{iz}-e^{-iz}}{2i}$$ so you want to see whether the equation $$e^{2iz}-2iwe^{iz}-1=0$$ has a solution for every $w\in\mathbb{C}$. Since $0$ is not a root of $t^2-2iwt-1=0$, you are done, because $\exp$ only omits the value $0$.
For the tangent, you want to solve $$\frac{e^{2iz}-1}{e^{2iz}+1}=iw$$ that is $$(1-iw)e^{2iz}=1+iw$$ which is solvable provided $iw\ne-1$ and $iw\ne1$, that is $w\ne i$ and $w\ne-i$.
Little Picard Theorem: If a function $f : \mathbb C \to \mathbb C$ is entire and non-constant, then the set of values that $f(z)$ assumes is either the whole complex plane or the plane minus a single point.