PLease HELP with this trigonometric equation

AI Thread Summary
The discussion revolves around solving the trigonometric equation tan(10xπ) = -10πx, which is believed to have infinite solutions within the interval x ∈ [-1, 2]. The author of the book suggests specific solutions in the form of xi = (2i-1)/20 + Ei for positive integers and xi = (2i+1)/20 + Ei for negative integers. There is confusion regarding how these particular values of xi are derived, especially since the equation is not defined for all real numbers. Participants express uncertainty about the infinite nature of solutions and seek clarification on the derivation of these values. The conversation highlights the complexities of trigonometric equations and the need for a deeper understanding of their solutions within specified domains.
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
 
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