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