I'm preparing for the exam which I will have next week and in one exercise I was asked to sketch a graph of $y=\sin(\sin(x))$. I didn't know how to deal with it. Now I know how it should look like but do you have any methods how to imagine it and, then, to sketch it? Not only this one but also e.g. $y=\sin(\cos(x))$.
I will be really grateful for some hints.
On
Something you could try is graph several different examples to see how each one is different. For example, graph $\sin(x)$ then graph $\sin(\sin(x))$. You will see it is the same graph only $\sin(\sin(x))$ has a slightly smaller amplitude and can't reach $1$ or $-1$ in the y axis. If you graph $\cos(x)$ and $\sin(\cos(x))$, $\sin(\cos(x))$ will be the same graph with a smaller amplitude and shifted up but similar to the $\cos(x)$ graph. So just see how each graph relates for each example.
Well, $\sin$ has range $[-1,1]$. So you're applying $\sin$ to something between $-1$ and $1$, so you need to first know how $\sin$ looks like on $[-1,1]$. It's increasing on this range. So from $-\pi/2$ to $\pi/2$, $\sin(\sin(x))$ is increasing from $-\sin(1)$ to $\sin(1)$. Note that $1$ is slightly less than $\pi/3$, so $\sin(1)$ is a bit less than $\frac{\sqrt{3}}{2}$. If you already know that $\sqrt{3}$ is about $1.7$, then you know $\sin(1)$ will be a bit less than $0.85$.
As you go to the right of $\pi/2$, $\sin(\sin(x))$ starts decreasing until $3 \pi/2$, then it starts increasing again, and so on. I'm not sure how much can be said about the actual shape of the graph without a calculator or some techniques from calculus.