How to solve the trigonometric equation sin(x) = x/2?

[prasannapakkiam]

How would one go about solving sin(x) = x/2
I.e. the intersections of
f(x)=sin(x)
&
g(x)=x/2

I can rigorously solve this by going to each individual period and finding the intersections. But is there a better way?

 

[arunbg]

You can find the no. of solutions easily enough using graphs, but getting the actual solution, would require rigour.

 

[prasannapakkiam]

This is going to do my scholarship exam on Friday. I was only notified today that I was selected. Although I knew most of the requirements prior to this, I did not know about trigonometric equations such as this. I cannot afford to draw graphs - time constraints are not going to do me favours. I know for sin(x)=k, there is a simple general solution rule. But as for kx; are there any quicker methods?

 

[Werg22]

Do you want to know the exact intersections or the number of intersections? If it's the number, it's fairly easy. The maxima and minima of sin x all have y = 1 and y = -1. The function x/2 is equal to 1 at x = 2 and - 1 at x = -2. Since the pi/2< 2 <pi, then it has to cross sin x at two points on the positive x-axis (picture this in your mind: the line has to cross the "mountain" between 0 and pi). Same applies to -2 > -pi. There are in total 3 intersection points (x = 0 is common to the positive and negative sides of the x axis).

 

[prasannapakkiam]

Yes I suppose knowing the number of solutions may be helpful. However, the values are also expected...

 

[Werg22]

Then, there exist no analytical method. Apart from the obvious x = 0 solution, the others have to be found by other method. Are you familiar with Newton's method?

 

[prasannapakkiam]

No I am not aware of Newton's method

 

[Werg22]

Then learn about it, you haven't got much time! Though, it's strange that they would ask you this kind of question...

 

[prasannapakkiam]

Wow! I have just learned it! It is quite accurate with just 3 steps. Since they expect 3 s.f. it is perfect. Thanks.