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

  • Context: Undergrad 
  • Thread starter Thread starter prasannapakkiam
  • Start date Start date
  • Tags Tags
    Trigonometric
Click For Summary

Discussion Overview

The discussion revolves around solving the trigonometric equation sin(x) = x/2, focusing on finding the intersections of the functions f(x) = sin(x) and g(x) = x/2. Participants explore various methods for determining the number of solutions and the actual intersection points, considering both graphical and analytical approaches.

Discussion Character

  • Homework-related
  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant suggests solving the equation by examining each individual period to find intersections, questioning if there is a more efficient method.
  • Another participant mentions that while the number of solutions can be determined graphically, finding the actual solutions requires more rigorous methods.
  • A participant expresses urgency due to an upcoming scholarship exam and seeks quicker methods for solving the equation without graphing.
  • One participant discusses the number of intersections, noting that the maxima and minima of sin(x) indicate there are three intersection points, including x = 0.
  • Another participant acknowledges the importance of knowing the actual values of the solutions, not just the number of intersections.
  • A participant states that there is no analytical method to find all solutions, except for the obvious x = 0, and inquires if others are familiar with Newton's method.
  • A later reply indicates a participant is unfamiliar with Newton's method but expresses a desire to learn it due to time constraints.
  • One participant shares their experience of quickly learning Newton's method and finds it effective for achieving the required significant figures.

Areas of Agreement / Disagreement

Participants generally agree that while the number of solutions can be estimated, finding the exact values requires additional methods. There is no consensus on a singular approach, as various methods are discussed and explored.

Contextual Notes

Participants mention time constraints and the need for quick solutions, indicating that some assumptions about familiarity with methods like Newton's may vary among them.

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?
 
Mathematics news on Phys.org
You can find the no. of solutions easily enough using graphs, but getting the actual solution, would require rigour.
 
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?
 
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).
 
Yes I suppose knowing the number of solutions may be helpful. However, the values are also expected...
 
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?
 
No I am not aware of Newton's method
 
Then learn about it, you haven't got much time! Though, it's strange that they would ask you this kind of question...
 
Last edited:
Wow! I have just learned it! It is quite accurate with just 3 steps. Since they expect 3 s.f. it is perfect. Thanks.
 

Similar threads

High School Can 2(sinx) be called a 'trigonometric function'?
  • · Replies 28 ·
Replies
28
Views
3K
High School Trigonometric Identity involving sin()+cos()
  • · Replies 11 ·
Replies
11
Views
2K
MHB Finding Points of Intersection for Trigonometric Functions
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Determine ## \beta ## as a function of ##\theta## linkage
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Why is it important to convert angle units when using trigonometric functions?
  • · Replies 8 ·
Replies
8
Views
2K
High School How would I write this trig solution?
  • · Replies 19 ·
Replies
19
Views
3K
MHB Solving Trig Identity: Sin[1/2 sin^−1 (x)] = 1/2 X (sqr(1+x) –sqr(1-x))
  • · Replies 2 ·
Replies
2
Views
1K
MHB How to Solve the Trigonometric Equation 6*Sin^2(x) - 3*Sin^2(2x) + Cos^2(x) = 0?
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Why fixed point iteration of ##x^3 = 1-x^2## doesn't converge when ##x_0= 0##
  • · Replies 16 ·
Replies
16
Views
2K
Undergrad Transforming coordinates between Cartesian and spherical
  • · Replies 1 ·
Replies
1
Views
3K