PLease HELP with this trigonometric equation

[joanmanuelbl]

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... :yuck:

Thank you so much
JoanManuel

 

[Gib Z]

Welcome to Physicsforums.com!

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.

 

[joanmanuelbl]

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? :grumpy:
why is it 2i-1/20 or the other way?
Please help me

 

[joanmanuelbl]

here is the equation as an image

Greetings, here is the equation:
[img=http://aycu20.webshots.com/image/10219/2000193818725334174_rs.jpg]
Could somebody explain me, please, how the autor obtains both values of xi? :grumpy:
Please I really need your help, take in account that the domain goes from -1 to 2 and it is a sinusoidal function