PLease HELP with this trigonometric equation

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Discussion Overview

The discussion revolves around a trigonometric equation involving the tangent function, specifically the equation tan(10xπ) = -10πx, and seeks to understand the derivation of proposed solutions within the domain x ∈ [-1, 2]. Participants are exploring the nature of the solutions and the reasoning behind the author's conclusions regarding their form.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • JoanManuel states that the equation has infinite solutions but seeks clarification on how the author derived specific solutions of the form xi = (2i-1)/20 + Ei and xi = (2i+1)/20 + Ei.
  • Another participant questions the assumption of infinite solutions, suggesting that the range of u derived from the transformation u = -10πx limits the number of solutions to the equation tan(-u) = u.
  • JoanManuel expresses confusion about the specific values of xi, asking for an explanation of the terms (2i-1)/20 and (2i+1)/20.
  • A later post includes an image of the equation, indicating a desire for visual clarification on the author's reasoning.

Areas of Agreement / Disagreement

Participants do not appear to reach consensus on the nature of the solutions, with some asserting the existence of infinite solutions while others challenge this view based on the defined range of the variable.

Contextual Notes

The discussion highlights the dependence on the definitions of the variables involved and the constraints imposed by the domain of the equation. There are unresolved questions regarding the mathematical steps leading to the proposed solutions.

joanmanuelbl
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Greetings my friends:
I have been reading a book about optimization and I found the following trigonometric equation:
tan(10x.pi)= - 10 pi x (this equation goes from x E [-1,2]
it is easy to see that has infinite solutions, but the author came to the conclusion that the solutions are:
xi=(2i-1)/20+Ei, for i=1,2,...

x0=0

xi=(2i+1)/20+Ei, for i=-1,-2,...


HOw does he get those probable solutions, please I really need to know...

Thank you so much
JoanManuel
 
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I may just be sleepy, but I can't see why its so obvious that there are an infinite number of solutions...let u=-10 pi x.

we want solutions to tan(-u)=u, or -tan (u)=u. Since u=-10 pi x, and x E [-1,2], u E [-20pi, 10 pi]. It would have an infinite number of solutions if u E all R, but that is not the case.
 
thank for the reply but I still have doubts

Thank you for the reply, I will put the equation in a better wayat I do not know from where the author obtains the values of xi?
why is it 2i-1/20 or the other way?
Please help me
 
Last edited:

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