An equation f is given by f (x) = (1 / x) - (cos (x) / (sin (x)). Prove that the equation f (x) = 0 has no solutions in (0, π) and has 1 solution in the interval (π, 2π).

Question

frostbite · Answer

The proof most be shown mathematical and not graphical.

nipunmalhotra93 · Answer

f(x)=0 will simplify to tanx=x.
You can show that this holds for no x in 0 to pi/2 and pi/2 to pi using taylor expansion.

frostbite · Answer

Emm i haven't learned about taylor expansions and Taylor polynomials. so there you lost me

frostbite · Answer

or at least not yet.

nipunmalhotra93 · Answer

actually, to express trigonometric functions in terms of polynomials, you need taylor series. As x is a polynomial, you you need taylor series to express tanx in terms of x.

nipunmalhotra93 · Answer

you can only do it graphically then.

frostbite · Answer

Any chance you can give a 10 sec couse? ^^

frostbite · Answer

*course

nipunmalhotra93 · Answer

ask anything you want.... :)

nipunmalhotra93 · Answer

you can give the graphical stuff an analytical treatment too....

frostbite · Answer

Emm... what to do, now i just fasted looked up in my book about Taylor polynomials of trigonometric functions, but i don't see the connection in what we want to show

nipunmalhotra93 · Answer

see... when you have any trigonometric function f(x), you can do absolutely nothing to solve an equation like f(x)=P(x), where P(x) is a polynomial. Solving means finding an x that satisfies this equation. Clearly, in our case, f(x)= tanx and P(x) is x. So, using taylor series, you can find an x that is the solution to this equation (if it exists)

frostbite · Answer

I see, but what would our taylor series be then?

nipunmalhotra93 · Answer

Actually, taylor's thm is more complicated than you think it is. It is true for the function in only an interval (you won't be able to understand the condition now). It'd be better if you study it first. You'll have to expand tanx about x=0 first. Ant then, you'll have to expand it about x=pi.

nipunmalhotra93 · Answer

It'd make much sense if you try to understand this question without perfectly understanding the theorem...

nipunmalhotra93 · Answer

The best thing you can do to solve this problem analytically is to do a graphical analysis.
This way, you will still solve the question... but without drawing the graph.