Sign of Secant
Jump to navigation
Jump to search
Theorem
Let $x$ be a real number.
\(\ds \sec x\) | \(>\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | |||||||||||
\(\ds \sec x\) | \(<\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ |
where $\sec$ is the real secant function.
Proof
For the first part:
\(\ds \cos x\) | \(>\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | \(\quad\) Sign of Cosine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 1 {\cos x}\) | \(>\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | \(\quad\) Reciprocal of Strictly Positive Real Number is Strictly Positive | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \sec x\) | \(>\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | \(\quad\) Secant is Reciprocal of Cosine |
For the second part:
\(\ds \cos x\) | \(<\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ | \(\quad\) Sign of Cosine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 1 {\cos x}\) | \(<\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ | \(\quad\) Reciprocal of Strictly Negative Real Number is Strictly Negative | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \sec x\) | \(<\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ | \(\quad\) Secant is Reciprocal of Cosine |
$\blacksquare$