# Graphs of funtions such as $y=\sin(\sin(x))$

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.

• Is a normal calculator available at the exam? – wythagoras Oct 31 '15 at 17:14
• If you mean that one with only 4 basic operations. Yes, it is. – Michael Oct 31 '15 at 17:31
• Does it not have the trig functions? You could always use taylor series to estimate some values and sketch. – YoTengoUnLCD Oct 31 '15 at 17:33
• Taylor series for this? That's not a smart idea as it will not be possible to even find by hand when the function is zero or has local extrema. Much better to use the kind of logic Ian lays out below – Simon S Oct 31 '15 at 17:38
• this is fun wolframalpha.com/input/… – janmarqz Oct 31 '15 at 17:53

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.
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.