Can You Prove This Trigonometric Inequality?

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SUMMARY

The discussion centers on proving the trigonometric inequality $\tan 6^{\circ}+\tan 24^{\circ}+\tan 42^{\circ}+\tan 86^{\circ} > 4$. Participants engage in verifying the correctness of the proof, with one user acknowledging and correcting mistakes pointed out by another. The focus is on the specific angles and their tangent values, which are critical to establishing the validity of the inequality.

PREREQUISITES
  • Understanding of trigonometric functions, specifically tangent.
  • Familiarity with angle measures in degrees.
  • Basic knowledge of inequalities in mathematics.
  • Experience with mathematical proof techniques.
NEXT STEPS
  • Research the properties of tangent functions at specific angles.
  • Explore methods for proving inequalities in trigonometry.
  • Study the behavior of tangent functions near critical angles.
  • Learn about mathematical proof verification techniques.
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Mathematics students, educators, and anyone interested in trigonometric proofs and inequalities.

anemone
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Prove $\tan 6^{\circ}+\tan 24^{\circ}+\tan 42^{\circ}+\tan 86^{\circ}\gt 4$.
 
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anemone said:
Prove $\tan 6^{\circ}+\tan 24^{\circ}+\tan 42^{\circ}+\tan 86^{\circ}\gt 4$.

My solution:

First we know

[TABLE="class: grid, width: 700"]
[TR]
[TD]$\tan x=x+\dfrac{x^3}{3}+\dfrac{2x^5}{15}+\cdots$[/TD]
[TD]$86^{\circ}=\dfrac{86^{\circ}\pi}{180^{\circ}}>1.5$ rad[/TD]
[TD]$42^{\circ}=\dfrac{42^{\circ}\pi}{180^{\circ}}>0.733$ rad[/TD]
[/TR]
[/TABLE]

So we have

$\begin{align*}\tan 42^{\circ}+\tan 86^{\circ}&\gt (1.5+\dfrac{1.5^3}{3}+\dfrac{2(1.5)^5}{15}+\cdots)+(0.733+\cdots)\\&>3.6375+0.733\\&>4.3705\end{align*}$

Since $\tan 6^{\circ},\, \tan 24^{\circ}>0$ we can conclude by now $\tan 6^{\circ}+\tan 24^{\circ}+\tan 42^{\circ}+\tan 86^{\circ}\gt 4$ and we're hence done.
 
Last edited:
anemone said:
My solution:

First we know

[TABLE="class: grid, width: 700"]
[TR]
[TD]$\tan x=x+\dfrac{x^3}{3}+\dfrac{2x^5}{15}+\cdots$[/TD]
[TD]$86^{\circ}=\dfrac{86^{\circ}\pi}{180^{\circ}}<1.5$ rad[/TD]
[TD]$42^{\circ}=\dfrac{42^{\circ}\pi}{180^{\circ}}<0.733$ rad[/TD]
[/TR]
[/TABLE]

So we have

$\begin{align*}\tan 42^{\circ}+\tan 86^{\circ}&\gt (1.5+\dfrac{1.5^3}{3}+\dfrac{2(1.5)^5}{15}+\cdots)+(0.733+\cdots)\\&>3.6375+0.733\\&>4.3705\end{align*}$

Since $\tan 6^{\circ},\, \tan 24^{\circ}>0$ we can conclude by now $\tan 6^{\circ}+\tan 24^{\circ}+\tan 42^{\circ}+\tan 86^{\circ}\gt 4$ and we're hence done.

should be

$86^{\circ}=\dfrac{86^{\circ}\pi}{180^{\circ}}>1.5$ rad
 
Thanks kaliprasad for catching it, I just fixed the mistakes.:o
 

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