Trigonometric Identity?

Question

Is there a trig identity that solves this equation?

x - sin x = 1.5

The problem is in radians. The answer is 2.27 radians, but I have no idea how that was obtained. Other resources say to graph it and find the x intercepts. Is there a trig identity that will solve this problem?

Answer 1

I am not aware of any identity which will help solve this problem.

A graph will give you an approximate root and if your graphing calculator is sophisticated, it will solve the problem for you. Mine gave 2.267172.

However, there are 3 additional comments which may be helpful.

If you expand sin in a Taylor's series, the first approximation to a root is x^3 = 90, i.e. x = 2.08.

Newton's method can be used (if you don't know the formula, look it up!). I used 2.08 as a first approximation and got 2.227 as my 2nd approximation.

Lastly, you can also use interval halving starting from 2 pi/3 and 3pi/4 which you can see from a graph.