Can You Prove This Trigonometric Inequality?

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Discussion Overview

The discussion revolves around proving the trigonometric inequality $\tan 6^{\circ}+\tan 24^{\circ}+\tan 42^{\circ}+\tan 86^{\circ}\gt 4$. Participants share their approaches and solutions related to this inequality.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant presents the inequality to be proven, stating it as a problem to solve.
  • Another participant reiterates the same inequality, indicating they have a solution to propose.
  • A subsequent post indicates that the participant has a solution but does not provide details in the quoted text.
  • A later reply acknowledges a correction made to the solution, thanking another participant for identifying mistakes.

Areas of Agreement / Disagreement

The discussion does not show clear consensus, as multiple participants are presenting their solutions and corrections without resolving the inequality or agreeing on a single approach.

Contextual Notes

There are indications of mistakes in the proposed solutions, but the specific nature of these mistakes and their implications remain unresolved.

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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