Do there exist non trivial solutions of the equation tan x = x and if they do exist, how to evaluate them algebraically? I graphed it on Desmos and got many random - looking solutions, is there any specific pattern or equation which describes their occurrence and if it does, what is the nature of these numbers? I mean like rational, irrational, transcendental etc
124065
I know that the equation $$ \tan(x) = x $$ can be solved using numerical methods, but I’m looking for a closed form of the solutions. In my opinion, having only numerical solutions means that we don’t know the problem, and sooner or later, we’ll be able to find a closed-form solution or at least a power-series solution. I’m looking for an explicit form of a sequence $ (x_{n})_{n \in \mathbb{Z}} $ in $ \mathbb{R} $ such that  $ 0 &lt; x_{0} &lt; \dfrac{\pi}{2} $, $ \tan(x_{n}) = x_{n} $ for all $ n \in \mathbb{Z} $, and $ (2 n - 1) \dfrac{\pi}{2} &lt; x_{n} &lt; (2 n + 1) \dfrac{\pi}{2} $ for all $ n \in \mathbb{Z} \setminus \{ 0 \} $.  The existence of such a sequence follows from the continuity of $ x \mapsto \tan(x) - x $ over its domain. Its uniqueness follows from the monotonicity of $ \tan(x) - x $ over the intervals $$ \left( (2 n - 1) \frac{\pi}{2},(2 n + 1) \frac{\pi}{2} \right), \quad n \in \mathbb{Z}. $$
106601
I understand that the quadratic equation can solve any second order polynomial. Furthermore, equations exist for polynomials up to fourth order. However, without a graduate level degree and a deep understanding of mathematics, is there an explanation of how we can prove that the equations for solutions to a fifth order and above don't exist?  Thanks in advance.
107848
How to find this ?   I'm stuck at $1$ and how to $1$ to change the tangent associated with a,b or c.  I meant this ,everybody.  $\frac{tan(a)+tan(b)+tan(y)−tan(a)tan(b)tan(y)}{1−tan(a)tan(b)−tan(a)tan(y)−tan(b)tan(y)}$
29601
This question concerns the equation $$\frac{a}{a^2+1} + \frac{b}{b^2+1} = \frac{c}{c^2+1}$$ and the possibility of rational number solutions with $abc \ne 0$.  In comments arising from:  It is mentioned that this equation has an infinite number of such solutions.  How can we determine that there are an infinite number of such solutions?  And is there a way to generate these solutions (or at least find a few of them)?
56483
$A=\begin{pmatrix}0&amp;0&amp;1\\ 0&amp;1&amp;0\\ 1&amp;0&amp;0\end{pmatrix}$  How do I show that there are no solutions for $X^2=A$ in $M_3(R)$?  I tried $X^2=\begin{pmatrix}a_{11}&amp;a_{12}&amp;a_{13}\\ a_{21}&amp;a_{22}&amp;a_{23}\\ a_{31}&amp;a_{32}&amp;a_{33}\end{pmatrix}^2=A$, but I couldn't seem to find anything that shows there are no solutions. How should I approach this?
9389
This came up in proving non-regularity of a certain language (powers of 2 over the ternary alphabet). Any clue to the above equation could help me move forward.  Edit: Of course, $x = 1, y = 1$ is a solution. I am looking for non-trivial solutions.
75098
I understand how to find the general solution of a differential equation, for complex roots, simple roots, nonhomogeneous equations etc... but I still don't really know what the general solution is?   As in, what is the purpose of it? What is it used for? thanks...
77357
This problem has baffled me. I have tried using $t = \tan(x/2)$ identities but end up with a 4 th order polynomial I cant solve.
116770
I know that if $x$ is a rational multiple of $\pi$, then $\tan(x)$ is .  Is there a fairly simple way to express $x$ as $\pi\frac{m}{n}$, if $\tan(x)$ is given as a square root of a rational?
6795
I learnt that for the equation: $Ax=b$   There is one solution if $A$ is invertible.  But if $A$ is singular there are infinity solutions or no solutions at all.  If $A$ is singular is it possible to determine whether there are no solutions or infinity solutions?
63201
Find all real solutions to the equation:$$4^x+6^{x^2}=5^x+5^{x^2}$$  Obviously $0$ and $1$ satisfy this equation, but here $x$ is a real number, and I do not know how to do the next step, in fact, I guess this equation is not in addition to $0$ and $1$ other than the root. But you know, I need a proof! Could anyone help me? Thanks a lot!
69928
Given the following problem:     Determine (without solving the problem) an interval in which the solution is certain to exist for the initial value problem $y' + (\tan t)y = \sin t, \space y(2\pi) = 0$.   I was able to get the right answer, but I do lack the understanding as to why it is the right answer. I simply took $2\pi$, and added and extracted $\frac{\pi}{2}$ to get $[\frac{3\pi}{2},\frac{5\pi}{2}$]. Seeing as it is the right answer, I understand that both $\tan t$ and $y(2\pi) = 0$ are essential parts, but the text does not explain this properly.
105231
What solutions does the equation $4x\equiv2\mod10$ have? (Hint, it will have more that one.) What about solutions to an equation $ax\equiv d\mod n$, where $d=\gcd(a,n)$?  This is the question that appeared in our universities last year paper. But it seems very difficult for me like for first part I am able to solve by simplifications I got $x=5\cdot Z +3$ as solution which makes sense and seems fair but for the second one, I have no idea how can I approach that. Can you please help me out with that one. I tried to simplify that by using Euclidean algorithm but still, I didn't get any idea that what actually the solution will be. My final exam is on 2nd of next month and because of covid I am unable to contact in person for any help. So any help from this site will be appreciated. Thank you for looking at this question!
105972
Solve the following equations for $x, (x ∈ N^+)$  $$ (a)\ 2φ(x) = x;$$ $$(b)\ 3φ(x) = x;$$ $$(c)\ 4φ(x) = x.$$  Can anyone help me with this, or give me some hint?
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.
105462
I am stuck the following problem :  Solve the following problem: $(1-\tan^2x)\sec^2x+2^{\tan^2x}=0$  Taking $p=\tan^2x,$ we get $(1-p^2)+2^p=0$. Now,I do not know how to go from here. Simple observation says $x=2n\pi \pm \frac{\pi}{3}$,satisfies the equation and so it will be the solution but I do not know how to get there. Can someone help?
21269
Can someone give me a hint on how can I solve the equation $$x^3 - x - 1 =0?$$  Thank you!