Answer
In general, $\sin{(bx)} \ne b\cdot \sin{x}$..
This is because they have different periods and amplitudes.
Refer to the image in the step by step part below for the graph.
Work Step by Step
RECALL:
The function $a \cdot \sin{(bx)}$ has :
amplitude = $|a|$
period = $\frac{2\pi}{b}$
Thus:
The function $y=\sin{(2x)}$ has an amplitude of $|1|=1$ and a period of $\frac{2\pi}{2} = \pi$.
The function $y=2\sin{x}$ has an amplitude of $|2|=2$ and a period of $\frac{2\pi}{1}=2\pi$.
From the information above, it is bvious that the t wo functions are different from each other.
Thus, it cannot be said that in general, $\sin{(bx)}=b \cdot \sin{x}$.
Use a graphing utility to graph the two functions.
(Refer to the attached image below for the graph, the green graph is $y=2\sin{x}$ while the red graph is $y=2\sin{x}$.)
Use a graphing utility to graph the given functions.
(Refer to the graph below.)
Notice that the graphs are different.
