# Prove that the equation $sin(x) = ax + b$ has at least one real root

I came across a question earlier this day, that I did not manage to solve. I have been asked to prove that the equation $\sin(x) = ax + b$ has at least one real root, for all $a, b$, where:

1) $a$ is not zero.

2) $b$ is a real number.

I have tried using the Intermediate Value Theorem in order to solve the problem, but without much luck.

Hoping for some help here, Thanks in advance!

• What function(s) have you tried when applying the IVT? – pjs36 Jan 31 '16 at 18:36

Hint: the function $f$ defined by $f(x) = ax + b - \sin x$ is continuous, and has limits $-\infty$ and $+\infty$ respectively at $-\infty$ and $+\infty$ (if $a > 0$; the other way if $a < 0$).
Apply the IVT on $f$.