I have an exercise that says:  (a) Prove that $\cos(z)$ and $\sin(z)$ are surjective functions from $\mathbb C \to \mathbb C$.  (b) Find the solutions of the equation $\cos(z)=\dfrac{5}{4}$.  As far as part (a) goes, I have no idea how to show surjectivity. I know by definition that $\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2}$ and $\sin(z)=\dfrac{e^{iz}-e^{-iz}}{2i}$. I want to show that given $w \in \mathbb C$, there are $z_1, z_2 \in \mathbb C$ such that $\cos(z_1)=w$ and $\sin(z_2)=w$. I've tried to play with the expressions from above but I couldn't get to anything. I need help on this.  For part (b), once I've proved (a), then I know (b) makes sense, i.e., there exist solutions for that equation.  $\cos(z)=\dfrac{5}{4}$ iff $\dfrac{e^{iz}+e^{-iz}}{2}=\dfrac{5}{4}$ Again, I have no idea what can I get out from this equation.
I am trying to solve the following two problems:  1) Prove that the functions $\cos(z)$, $\sin(z)$ are surjective over the complex numbers.  2) Find all $z \in \mathbb C$: $cos(z) \in \mathbb R$ and find all $z \in \mathbb C$: $\sin(z) \in \mathbb R$.  For 1),I've tried to prove it for the function $\cos(z)$ (I suppose the other one is analogue)  so I've used the fact that $\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2}$.  Let $w \in \mathbb C$, I want to show there exists $z \in \mathbb C : f(z)=w$, i.e., $\dfrac{e^{iz}+e^{-iz}}{2}=w$, multiplying by $2$ and then by $e^{iz}$ yields $e^{2iz}+1=2we^{iz}$ iff $e^{2iz}-2we^{iz}+1=0$.   I don't know if this approach is the correct one but here I've replaced $e^{iz}$ by $x$, so the solutions of the equation would be the roots of the polynomial $p(x)=x^2-2wx+1$, by the quadratic formula, I get that $x \in \{\dfrac{2w+w_0}{2},\dfrac{2w-w_0}{2}\}$, where $w_0^2=4w^2-4$.  Then, $e^{iz} \in \{\dfrac{2w+w_0}{2},\dfrac{2w-w_0}{2}\}$. At this point I got lost, I would like to explicitly show that $z$ exists and I can't see existence directly from the fact that $e^{iz}=\dfrac{2w+w_0}{2}$ or $e^{iz}=\dfrac{2w-w_0}{2}$.  First I thought of taking logarithm of both sides of the equation ir order to solve for $z$, but this is not a legitimate operation unless $e^{iz} \in \mathbb R$.   I couldn't go any farther.  For point 2) I have no idea what to do, should I use the identity $\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2}$?  I would appreciate some help with the two points (specially with point 2), at least in 1) I could do something).   Btw, Happy new year!
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The same thing happened when I recently flagged a post as not an answer.  Are the moderators being more strict when it comes to handling quality related flags? If so, why? Was Elrond, in Tolkien's opinion, more inherently powerful than Galadriel? I always thought it was the other way around, but then I read this (Letter 246):     "Of the others only Gandalf might be expected to master him – being an emissary of the Powers and a creature of the same order, an immortal spirit taking a visible physical form. In the 'Mirror of Galadriel', 1381, it appears that Galadriel conceived of herself as capable of wielding the Ring and supplanting the Dark Lord. If so, so also were the other guardians of the Three, especially Elrond. But this is another matter. It was part of the essential deceit of the Ring to fill minds with imaginations of supreme power."   This "especially Elrond" confuses me. I don´t think that Elrond is inherently more powerful than Galadriel, who is described as the greatest of the Noldor, except for Fëanor (and possibly even his equal).  A quote from Unfinished Tales says:     "...In this he (Tolkien) emphasized the commanding stature of   Galadriel already in Valinor, the equal if unlike endowments of   Fëanor; and it is said here that so far from joining in Fëanor's   revolt she was in every way opposed to him."   Furthermore,      "A queen she was of the woodland Elves, the wife of Celeborn of   Doriath, yet she herself was of the Noldor and remember the Day before   days in Valinor, and she was the mightiest and fairest (so even   fairer than Arwen I would guess) of all the Elves that remained in   Middle-earth."   (The Silmarillion, of the Rings of Power)  And again...     "Galadriel was the greatest of the Noldor, except Fëanor maybe, though   she was wiser than he, and her wisdom increased with the long years."   (the Shibboleth of Fëanor)  Restating it:     "These two kinsfolk (Fëanor and Galadriel), the greatest of the   Eldar in Valinor, were unfriends for ever." "Who together with the   greatest of all the Eldar, Luthien Tinuviel, daughter of Elu Thingol,   are the chief matter of the legends and histories of the Elves."   (Unfinished Tales)  The latter means that Luthien, Fëanor and Galadriel are the chief matter of the history and legends. It seems to me that Tolkien thought of those three as the creme de la creme of the elven crop.  Reading all that I have serious problems believing that Elrond could be superior to her. As great as Elrond is, descending from Melian, he is no equal of Fëanor, who is said to be the mightiest of all elves, and if Galadriel is his equal, then there is no way that Elrond would be mightier than her.  Maybe I just read this "especially Elrond" wrong; it could as well mean that especially Elrond was deceived into thinking that he could master the ring, because that was the deceit of the ring. And Tolkien said explicitly that only Gandalf could possibly master it.  So, what do you think? QGIS 3.0 and Qt 5.7 time schedule When will QGIS 3.0 with QT5.7 (possibly) be released as a stable and long term support version? "This is considered..." vs. "This is considered as..." Does this "as" change the meaning or correctness of this sentence?     This is considered as socially desired.   This is considered socially desired.   Are both of them grammatically correct? Why or why not? releasing a second question concerning ordered stuff and set theory, which is very similar to my first question () which I'm putting here for reference.  The next problem that I'm stuck at in my book is this: we have an ordered triple $(a,b,c)$ and we want to define it using set theory. 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