Sign of Cotangent
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Theorem
Let $x$ be a real number.
Then:
| \(\ds \cot x\) | \(>\) | \(\ds 0\) | if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \cot x\) | \(<\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$ |
where $\cot$ is the real cotangent function.
Proof
For the first part:
| \(\ds \tan x\) | \(>\) | \(\ds 0\) | if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$ | \(\quad\) Sign of Tangent | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac 1 \tan x\) | \(>\) | \(\ds 0\) | if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$ | \(\quad\) Reciprocal of Strictly Positive Real Number is Strictly Positive | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cot x\) | \(>\) | \(\ds 0\) | if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$ | \(\quad\) Cotangent is Reciprocal of Tangent |
For the second part:
| \(\ds \tan x\) | \(<\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$ | \(\quad\) Sign of Tangent | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac 1 {\tan x}\) | \(<\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$ | \(\quad\) Reciprocal of Strictly Negative Real Number is Strictly Negative | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cot x\) | \(<\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$ | \(\quad\) Cotangent is Reciprocal of Tangent |
$\blacksquare$