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!
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 $f$.