Subject: Trig question

If x is in radians, how do you solve for x, where: x-sin(x) = constant?

Keith Roble
secondary
other (professional)

I know of no way to directly solve this equation, the best you can do is approximate a solution. The most widely used technique is Newton's method, or sometimes called the Newton-Raphson method. This technique applies to equations of the form f(x) = 0 where f(x) is a differentiable function and requires the function f(x) and its derivative f'(x). The procedure progresses through a seuqence of steps.

• First you need a initial approximation, x₁ to the solution
  
  
• Calculate a second approximation using the expression
  
  
• Calculate a third approximation using the expression
  
  
• In general, at each step calculate the next approximation using the expression
  
  

In many situations the sequence x₁, x₂, x₃, ... will converge to a solution of the equation f(x) = 0.

For your problem suppose that the constant is C and then the equation x-sin(x) = constant can be written f(x) = 0 where

Here, f'(x) = -1 + cos(x) and the iterative step is given by

I tried this technique with C = 10 and x₁ = 10 with the following results

You need be careful when applying Newton's method. In particular it is sensitive to the initial approximation x₁. For some values of x₁ the sequence may not converge, or may converge to a solution which is different from the solution you are seeking. I suggest that you find a reference on Newton's method. You can find it in most Calculus books.