Solve the equation x - sinx = 0.
3 Answers | Add Yours
This is a transcedental equation and for solving it we have to differentiate the function.
The derivative is a monotone increasing function
( -1<cosx<1), so the difference 1 -cos x>0 =>f(x)>0, so f(x) is an injection.
We'll calculate f(0):
Because f(x) is an injection, that means that f(x) is increasing, so, x=0 is the only solution for x-sinx=0.
To solve x-sinx = 0.
We know that sinx < x always for all x
So sinx /x < 1 always.
But Lt x--> 0 sinx/x = 1 is a well known limit.
Therefore sinx/x = 1 has no other solution than x=0.
==> sinx = x
If you assume that there are two functions:
y=sinx and y=x
The graph will show that both functions only intersect at one point (0,0)
Then the only solution is x=0
sin 0 = 0
The only value for the equality is 0
sin 0 = 0
x-sinx > 0 for x>0
x-sinx <0 for x<0
x-sinx=0 for x=0
Join to answer this question
Join a community of thousands of dedicated teachers and students.Join eNotes