Find the period of the trigonometrics functions $\sin[\cos(x)]$ and $\cos[\sin(x)]$

Question

So I need to find the period of these two trigonometric functions:

$$\cos[\sin(x)]$$ and $$\sin[\cos(x)]$$

but algebraically and without using the graph of this functions.

So to be more clear, my question is: Is there any algebraic method or any mathematical trick I can use it to find the periods of these two trigonometric functions?' even without the proof.

Like the period of $\cos(bx)$ is $2\pi/b$ directly without drawing the graph and without calculations.

A function is periodic if $f(x)=f(x+T)$ for some value of $T$ — Doobius, May 22 '22 at 18:01
Look up "composition of periodic functions". This will give you some insight. Iff you don't find anything useful, I will see if I can write an answer about it sometime. This is relevant, but it doesn't show that $\pi$ is the period (rather, just that the composition is periodic). — Sarvesh Ravichandran Iyer, May 25 '22 at 04:48

score 5 · Answer 1 · answered May 22 '22 at 18:02

It is clear that $f : x \mapsto \cos(\sin(x))$ is periodic, since $f(x+2\pi)=f(x)$ for every $x$. Let's find its period.

Let $T \in \mathbb{R}$. One has \begin{align*} & \forall x \in \mathbb{R}, \quad f(x+T)=f(x) \\ \Longleftrightarrow \quad & \forall x \in \mathbb{R}, \quad \cos(\sin(x+T))=\cos(\sin(x))\\ \Longleftrightarrow \quad & \forall x \in \mathbb{R}, \quad |\sin(x+T)|=|\sin(x)| \\ & \quad \quad \quad \quad \quad \quad \textit{(because } \sin(x),\ \sin(x+T) \in [-1,1])\\ \Longleftrightarrow \quad & T \in \pi\mathbb{Z}\\ & \quad \quad \quad \quad \quad \quad \textit{(because } |\sin| \textit{ is } \pi-\textit{periodic})\\ \end{align*}

So the period of $f$ is $\pi$.

Try the same approach with $x \longmapsto \sin(\cos(x))$.

Find the period of the trigonometrics functions $\sin[\cos(x)]$ and $\cos[\sin(x)]$

1 Answers1