# Possible to solve without guessing/graphing?

• Apr 28th 2013, 06:39 AM
Latsabb
Possible to solve without guessing/graphing?
I was working a fairly simple problem, and then I hit a snag. I know the answer already, and I know that I could find it by plugging it into a graphing calculator, or by simply guessing, but I was hoping that there was some sort of way to solve it with pure mathematics.

cos(x)-xsin(x)=0

That ends up turning into x=cos(x)/sin(x) or x=cot(x).

Is there some way to solve for x, in this situation? Some special rule, or formula, or something? Like I said, I know that graphing it will give you a solution, and random number plugging will solve it as well, but I am hoping a different way. Thanks!
• Apr 28th 2013, 06:55 AM
HallsofIvy
Re: Possible to solve without guessing/graphing?
That equation involves x both "inside" and "outside" a transcendental function. There is no standard "algebraic" method of solving it and I suspect there is no "standard" notation for writing the number that is its solution. It reduces, of course, to x= cot(x). I'm not sure what you mean by "guessing" but graphing would be one way of getting an approximate solution. There are also methods for numerically getting approximate solutions to any desired accuracy, by "bisection" or "Newton's method", say.
• Apr 28th 2013, 07:03 AM
Latsabb
Re: Possible to solve without guessing/graphing?
By guessing, I meant just plugging in numbers for x. I plug in 1, get a value, and then try two, etc. When solving, the solution should get closer and closer to your guessed number as long as you increase or decrease consistently. When you see that they are no longer approaching each other, you have passed the answer, and can boil it down further. Basically just guessing X values (although not randomly) trying to find when the cotangent of X would give you X. It can be done, or closely approached by guessing, but is very time consuming. Graphing it would be much quicker, but I was hoping for a way to avoid drawing out a graph. (my exams dont allow the use of a graphing calculator) But if there isnt any way to do it algebraically, then my only option would be to make a graph, and approximate based on the graph.
• Apr 28th 2013, 09:59 AM
BobP
Re: Possible to solve without guessing/graphing?
Sketching a graph to obtain a first approximation to a root is fine and is usually the first thing to do, but if you are to find the root of this equation to any reasonable degree of accuracy, you are going to have to use a numerical method.
HallsofIvy has suggested two methods, there are others. Newton-Raphson requires a bit of calculus and is normally the method of choice. If you are not into calculus then it's Bisection or Regula Falsi (method of false position).