0

Let $a, y \in \mathbb{R}^+$ with $y<a$, and $b, x \in [0, \pi/2]$. Now consider the system of two equations given by

\begin{align} a &= \frac{1}{\tan(x)} + y\\ b & = (x+y) \mod \frac{\pi}{2} \end{align}

where mod with real numbers is defined in the natural way. Now, for fixed, known values $a, b$, I am supposed to determine if there exist an uncountably infinite, countably infinite, or finite set of $x, y$ pairs that satisfy the system of equations. Would anyone happen to know how to go about doing this, saying how they arrived at the answer they did? Thanks!

Greg Martin
  • 78,820
user918212
  • 507
  • Notice that if you change $x$ by an integer multiple of $\pi$, then neither right side changes. – Gerry Myerson Apr 13 '20 at 03:16
  • @GerryMyerson Indeed, however we have the requirement that $x \in [0, \pi/2]$ – user918212 Apr 13 '20 at 03:18
  • Sorry, I overlooked that. – Gerry Myerson Apr 13 '20 at 03:23
  • It seems that there is always one solution - the tan part provides a continuous vertical not line (the tan shape) and a line segment crossing it. – Moti Apr 13 '20 at 06:07
  • @Moti that is what I noticed too, however I am not sure if the mod allows for more than one solution, because you can rewrite the mod part as $b=(x+y)c+\pi/2 d$ where c and d are integer parameters for this line, so you can obtain multiple intersection points. – user918212 Apr 13 '20 at 14:12
  • You are right. There are parallel lines separated by $pi/2$ – Moti Apr 13 '20 at 20:55

0 Answers0