Sign of Cosine
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Theorem
Let $x$ be a real number.
Then:
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \cos x\) | \(<\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ |
where $\cos$ is the real cosine function.
Proof
Proof by induction:
Base case
For $n = 0$, it follows from Sine and Cosine are Periodic on Reals/Corollary.
Induction Hypothesis
This is our induction hypothesis:
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \cos x\) | \(<\) | \(\ds 0\) | for $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ |
Now we need to show true for $n=k+1$:
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n + \dfrac 3 2} \pi < x < \paren {2 n + \dfrac 5 2} \pi$ | |||||||||||
| \(\ds \cos x\) | \(<\) | \(\ds 0\) | for $\paren {2 n + \dfrac 5 2} \pi < x < \paren {2 n + \dfrac 7 2} \pi$ |
Induction Step
This is our induction step:
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n + \dfrac 3 2} \pi < x + 2 \pi < \paren {2 n + \dfrac 5 2} \pi$ | |||||||||||
| \(\ds \map \cos {x + 2 \pi}\) | \(>\) | \(\ds 0\) | for $\paren {2 n + \dfrac 3 2} \pi < x + 2 \pi < \paren {2 n + \dfrac 5 2} \pi$ | \(\quad\) Sine and Cosine are Periodic on Reals | ||||||||||
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n + \dfrac 3 2} \pi < x < \paren {2 n + \dfrac 5 2} \pi$ | \(\quad\) Replacing $x + 2 \pi$ by $x$ |
Also:
| \(\ds \cos x\) | \(<\) | \(\ds 0\) | for $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ | |||||||||||
| \(\ds \cos x\) | \(<\) | \(\ds 0\) | for $\paren {2 n + \dfrac 5 2} \pi < x + 2 \pi < \paren {2 n + \dfrac 7 2} \pi$ | |||||||||||
| \(\ds \map \cos {x + 2 \pi}\) | \(<\) | \(\ds 0\) | for $\paren {2 n + \dfrac 5 2} \pi < x + 2 \pi < \paren {2 n + \dfrac 7 2} \pi$ | \(\quad\) Sine and Cosine are Periodic on Reals | ||||||||||
| \(\ds \cos x\) | \(<\) | \(\ds 0\) | for $\paren {2 n + \dfrac 5 2} \pi < x < \paren {2 n + \dfrac 7 2} \pi$ | \(\quad\) Replacing $x + 2 \pi$ by $x$ |
The result follows by induction.
For negative $n$:
Induction Hypothesis
This is our induction hypothesis:
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \cos x\) | \(<\) | \(\ds 0\) | for $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ |
Now we need to show true for $n=k-1$:
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n - \dfrac 5 2} \pi < x < \paren {2 n - \dfrac 3 2} \pi$ | |||||||||||
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n - \dfrac 3 2} \pi < x < \paren {2 n - \dfrac 1 2} \pi$ |
Induction Step
This is our induction step:
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \map \cos {x + 2 \pi}\) | \(>\) | \(\ds 0\) | for $\paren {2 n - \dfrac 1 2} \pi < x + 2 \pi < \paren {2 n + \dfrac 1 2} \pi$ | \(\quad\) replacing $x$ by $x + 2 \pi$ | ||||||||||
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n - \dfrac 1 2} \pi < x + 2 \pi < \paren {2 n + \dfrac 1 2} \pi$ | \(\quad\) Sine and Cosine are Periodic on Reals | ||||||||||
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n - \dfrac 5 2} \pi < x < \paren {2 n - \dfrac 3 2} \pi$ |
Also:
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ | |||||||||||
| \(\ds \map \cos {x + 2 \pi}\) | \(>\) | \(\ds 0\) | for $\paren {2 n + \dfrac 1 2} \pi < x + 2 \pi < \paren {2 n + \dfrac 3 2} \pi$ | \(\quad\) replacing $x$ by $x + 2 \pi$ | ||||||||||
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n + \dfrac 1 2} \pi < x + 2 \pi < \paren {2 n + \dfrac 3 2} \pi$ | \(\quad\) Sine and Cosine are Periodic on Reals | ||||||||||
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | for $\paren {2 n - \dfrac 3 2} \pi < x < \paren {2 n - \dfrac 1 2} \pi$ |
The result follows by induction.
$\blacksquare$