The values of $a$ for which the function $f(x) = \sin x - ax + b$ increases on $R$ are ……...

The values of a for which the function $f(x) = \sin x - ax + b$

increases on $R$ are $( - \infty , - 1)$.
${f^\prime }(x) = \cos x - a$

and ${f^\prime }(x) > 0 \Rightarrow \cos x > a$

Since, $\cos x \in [ - 1,1]$

$\Rightarrow$ $a < - 1 \Rightarrow a \in ( - \infty , - 1)$