How many solutions?

Geometry Level 1

How many real numbers x x satisfy

π ⋅ sin ⁡ x = 2 x ? \pi \cdot \sin x= 2x?


The answer is 3.

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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5 solutions

Other than the trivial solution of x = 0 x=0 , we will find the others.

Rewrite equation as:

sin ⁡ x x = 2 π \frac { \sin { x } }{ x } = \frac { 2 }{ \pi }

It is easy to see that sin ⁡ x x \frac { \sin { x } }{ x } is a strictly increasing function in ( 0 , ∞ ) (0,\infty) and a strictly decreasing function in ( − ∞ , 0 ) (-\infty,0) . Hence sin ⁡ x x = 2 π \frac { \sin { x } }{ x } = \frac { 2 }{ \pi } is true for two values of x x , namely ( π 2 , − π 2 ) (\frac {\pi }{ 2},\frac { -\pi }{ 2 }) .

Therefore, the total number of solutions is: ( 0 , π 2 , − π 2 ) = 3 (0,\frac {\pi }{ 2},\frac { -\pi }{ 2 })\quad=\boxed{3}

9 Helpful 1 Interesting 1 Brilliant 1 Confused

Can you rather solve for those two values of x than just stating directly?

Vishal Yadav - 5 years, 2 months ago

Draw the graph.

6 Helpful 0 Interesting 0 Brilliant 0 Confused
Pranjal Gupta
Jul 18, 2015

see graph

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

How else can we approach this problem without drawing out the exact graph?

To the moderator There is a general approach that I have developed as there are many questions like this posted. So first step is to evaluate the range in which we can find solutions , and that can be easily found out , as the range of the sine function is from -1 to 1 . So (pi/2) *sin(x) will range from -pi/2 to pi/2,which means that we "can" find solutions in the range evaluated above. And then I treat the function 2x/pi . The graph is a straight line passing through the origin , which means it necessarily has 1 solution. Now if I make a guess by putting x=pi/2 and -pi/2 then I get intersection points. And that means that there will be no more solutions.

Hitesh Yadav - 11 months ago
Rahul Singh
Mar 4, 2015

It is an example of a transdental equation.can be easily solved by graphical method. let f(x)=pi.sinx and g(x)=2x.now draw their graphs seperately.it can easily be seen that the two graphs cut each other at 3 distinct points.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Adhiraj Dutta
Feb 23, 2020

Desmos Desmos

1 Helpful 0 Interesting 1 Brilliant 0 Confused

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