I have an equation,     $$\sinh z =z, \ z\in\mathbb{C}$$   How to solve it?  My question  It's trivial to see that the $z=0$ is a solution of the equation. But I want to know if there exists a non-zero solution.  The equation is from the answer of my another question.
91334
Show that $\sin(z) = z$ has infinitely many solutions in complex numbers. Little Picard theorem should help, but using big Picard theorem is undesirable. Thanks a lot!
75013
Suppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 &lt; x &lt; 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos x = 2$ where $0&lt;x&lt;2\pi$.  How do you solve this equation without using the method that moves $b\cos x$ to the right side and squaring left and right sides of the equation?  And how does solving $\sqrt{3}\sin x + \cos x = 2$ equal to solving $\sin (x+ \frac{\pi}{6}) = 1$
28096
How does one solve this equation?     $$\cos {x}+\sin {x}-1=0$$   I have no idea how to start it.  Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$?  Thanks in advance!
48783
Find all $z$ for the equation   $$(z+1)^7 = z^7$$    The different solutions can be unsimplified and both rectangular or exponential. I have the lead that I subsitute the $z+1$ term with a root of unity. Then, I got lost when I looked at that $z^7$. A thorough explanation would be great!
46357
I gave an exercise to a student the other day, thinking I had a simple solution for it, but it seems that my solution was just bullshit (at least, not conclusive). Well, at least, I warned her I wasn't sure of my solution :-)  Here it is : $a\in[-1,1]$, show that all roots of polynomial $P=X^{n+1}-aX^n+aX-1$ have modulus $1$ (I try different ways of expressing the problem, tell me which one is better english...).  It is easy to see that    roots of $P$ are $1$, $-1$ for some special cases, and complex not real roots $z_i$ such that $\overline{z_i}$ is also a root, as $P$ is kind of reciprocal, $z$ root implies $\frac1z$ is also a root$.   All the other things I tried are not conclusive. For example, $a\in[-1,1]$ may be translated as $a=\cos\theta$ for some $\theta\in\mathbb R$, but I don't see how to use this. I tried some rewriting, but nothing seems to work. I tried to work on the modulus of a root, tried the relations roots-coefficients...  I'm quite stuck here. Could you help me please ?  Thanks.
58995
Find all possible solutions of $$2^x + 3^y = z^2.$$    My approach.   First I substituted $x = 0$, and got the solution, then for $y = 0$. And for $x &gt;  0$ and $y &gt; 0$ , I just know the Pythagorean triple and got the solution as $(4,2,5)$ and $(4,2,-5)$. Please help me to solve it properly.
11545
In $(0 \:\:\pi)$Find the number of solutions of $$\sin x+2 \sin 2x- \sin 3x=3$$  The equation can be written as  $$\sin x+4 \sin x \cos x=3+\sin 3x$$ i.e.  $$\sin x(1+4\cos x)=3+\sin 3x$$ i.e.,  $$\sin x(1+4\cos x)=3+\sin x(3-4\sin^2 x)=3+\sin x(4\cos ^2x-1)$$  so  $$\sin x(4\cos^2 x-4\cos x-2)=-3$$ Any hint from here?
119571
I am investigating solutions to Fermat's equation $$x^n + y^n = z^n$$ with $x,y,z$ in the Gaussian integers, excluding integers and pure imaginaries.  I have found out that there are only trivial solutions for the $n=3$ and $n=4$ cases, e.g. .   I would be grateful if you let me know of the current status or if it is already a theorem.
3610
Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$  I have no idea how to deal with this crazy question. One idea came into my mine is factorization, but I can't go on... Can anyone help me please? Thank you.
82407
Suppose we have an equation $ab=0$. This equation is true when statements $a=0$ or $b=0$ are true.  If $a=0$, then $b=\frac{0}{0}$. That means $b$ could be any number for $ab=0$ to be true. If the set which groups all the numbers is the complex set, then $b$ will every number within $\mathbb{C}$, so $\forall z\in\mathbb{C}:b=\frac{0}{0}=z$. Therefore, $\frac{0}{0}$ is every number.  I know it really is not defined as number but conceptually it is every number, right?  Is this right, or am I missing something?
100189
Find the number of solutions to the equation $$2011^x+2012^x+2013^x-2014^x=0$$  The answer seems to be zero, but I have no idea why. Please avoid considering complex solutions and other scary things.
120317
How would I show the following.  For all odd integers a,b, and c if z is a solution of $ax^2+bx+c=0$ the z is not rational.  How would I show this? This is what I did  Let a be an odd integer then for a integer k it has the form $a=2k+1$  then $b=2t+1$ for if c integer.  then $c=2r+1$ if s is an integer then I do  $(2k+1)(x)^2+2t+1(x)+(2r+1)=0$  But I find my self stuck on what to do any hint are appreciated.
122307
Given $a,b,c,d \in \mathbb{Z}$, there is $x,y$ such that  $$(a^2 + b^2)(c^2 + d^2) = x^2 + y^2$$  One can show this by considering the complex number $a + bi$ and $c+ di$, using complex properties to deduce that $x = ac - bd, y = ad + bc$ is a solution.   However, given that either $a^2, b^2$ are distinct or $c^2,d^2$ are distinct, then how can one can find nonzero solutions $x,y$?  Sorry, I am assuming $a,b,c,d$ are nonzero.
91077
Just like if we have any quadratic equation which has complex roots, then we are not able to factorize it easily. So we apply quadratic formula and get the roots.  Similarly if we have a cubic equation which has two complex roots (which we know conjugate of each other) and one fractional root, then we are not able to find its first root by hit &amp; trial.  So my question is like quadratic formula, is there exist any thing like cubic formula which help in solving cubic equations?  For example, I have an equation $$2x^3+9x^2+9x-7=0\tag{1}$$ and I have to find its solution which I am not able to find because it has no integral solution. Its solutions are $\dfrac {1}{2}$, $\dfrac{-5\pm \sqrt{3}i}{2} $, I know these solutions because this equation is generated by myself.  So how can I solve equations like these?  Also while typing this question, I thought about the derivation of quadratic formula, which is derived by completing the square method.  So I tried to apply ‘completing the cube’ method on the general equation $ax^3+bx^2+cx+d=0$ but it didn't help.  So please help me in finding a cubic formula or to solve the equations like given in example by an alternative method.
8702
Find all roots in $\mathbb{C}$ of the equation $\cos(z)=2$ and plot them on an Argand diagram.  My solution: $\cos(z)=2$, $z=x+iy$ so we have that $\cos(x)\cos(iy)-\sin(x)\sin(iy)=2$ so $\cos(x)\cosh(y)=2$ and $\sin(x)\sinh(y)=0$ must hold. So if $\sinh(y)=0$ then $y=0$ so $\cosh(y)=1$ and so $\cos(x)=2$ which is a contradiction, therefore $x=n\pi$, for some $n\in \mathbb{Z}$ and so $\cos(x)=\pm 1$. However, $\cosh(y)&gt;0$, so if $\cos(x)=-1$ it follows that $\cos(x)\cosh(y)&lt;0$ and cannot equal $2$. Hence $\cos(x)=1$ so $x=2n\pi$ for some $n\in \mathbb{Z}$ and $y=\pm\cosh^{-1}(2)$ since $\cosh(y)=2$. Therefore since $z=x+iy$ $\implies z=2n\pi \pm i\cosh^{-1}(2)$, $n\in \mathbb{Z}$.  My question is, is this sufficent to answer the question $\cos(z)=2$? And if not how should I structure it using this method? Also why is it $\pm\cosh^{-1}(2)$? I don't understand this sutble point. And if I was to draw an Argand diagram would it just be on the imaginary axis at $\cosh^{-1}(2)$ every $2n\pi$?
38318
If $\alpha$ and $\beta$ are the solution of $$a \cdot \tan\theta + b \cdot \sec\theta = c$$, then show that $$ \tan(\alpha + \beta) = \frac{2ac}{a^2 - c^2}$$  I did the following:     $$a \cdot \tan\theta + b \cdot \sec\theta = c$$       Squaring both sides,  $$a^2 \cdot \tan^2\theta + b^2 \cdot \sec \theta + 2 \cdot a \cdot b \cdot \tan \theta \cdot \sec \theta = c^2$$      This implies $$(a^2 + b^2) \cdot \sin^2 \theta + 2 \cdot a \cdot b \cdot \sin \theta + (b^2 - c^2) = 0$$       Hence I get $$\sin \alpha + \sin \beta = \frac{-2ab}{a^2 + c^2}$$ and $$\sin \alpha \cdot \sin \beta = \frac{b^2 - c^2}{a^2 - c^2}$$   But I am not able to solve it further.
73339
I am unable to get a start for this question      $2x^2+3y^2=z^2$ does not have a non-zero integral solution.    Can anyone please give me some hint so that I can try this , as I am unable to start.  Thanks