How to write lies in quadrant $X$ symbolically?

Question

In Trigonometry, there are a lot of questions of the form :
Write all trigonometric ratios if $\cot x = \dfrac{12}{5}$ and x lies in quadrant III...

Is there some symbolic method to write that x lies in quadrant III? What about $x \in$ III?

Thanks!

Answer 1

One approach would be to write "angle $x$ is in quadrant III" as $x\in [\pi,3\pi/2]$. This notation is pretty common and would not require special definitions.

If one wants to capture the possibilities that angle $x$ "wraps around" one or more times, or goes negative to get into the third quadrant, we might write instead:

$$ x \in [\pi,3\pi/2] + 2\pi k \text{ for some } k \in \mathbb Z $$

On the other hand if one wishes to identify points that lie in the third quadrant, notice that these are exactly the points whose Cartesian are less than (or equal to) zero. So we might write:

$$ (s,t) \text{ where } s,t \le 0 $$

for points specifically in the third quadrant. A variation of this could be made to work for other quadrants.

Answer 2

Hint. Quadrant III is the set of all points $(x,y)\in \mathrm R^2$ such that $x\le 0$ and $y\le 0.$ Can you now continue?