Is that a known "constant"?

Question

I noticed, using a calculator, that the following operation:

$$Y = \sin(\cos(\tan(\log(n))))$$

for values of $n$ from $0.00000001$ to $99999999$, I obtain always quite the same number, id est

$$Y = 0.017452403 \pm 0.000000155$$

Is that a know "constant"?

Anyway the problem is also that when I plot the function

$$y = \sin(\cos(\tan(\log(x))))$$

I don't obtain a graph with the shape of a quite constant behavior. Indeed I obtain also negative values. What's wrong?

Answer 1

It certainly isn't constant - indeed, its range is $[\sin(-1), \sin(1)]$, which is noticeably larger than $[-{1\over 2}, {1\over 2}]$. So there is quite a lot of variation.

To see this, note that:

• the range of $\log(n)$ for $n\in (0.001, 1000)$ (say) strictly contains the interval $A=(-{\pi\over 2}, {\pi\over 2})$.

• The range of $\tan$ on $A$ is $(-\infty, \infty)=\mathbb{R}$.

• The range of $\cos$ on $\mathbb{R}$ is $[-1, 1]$.

• So the range of the whole function you're asking about on $(0.001, 1000)$ - or any other reasonably-large interval - is $[\sin(-1), \sin(1)]$.

Answer 2

You sample at points $x_k = k\, 10^{-8}$ for $k\in [1,10^{17}-10^8]$ this gives \begin{align} y_k &= \sin(\cos(\tan(\log(x_k)))) \\ &= \sin(\cos(\tan(\log(k)-8\log(10)))) \end{align} The three trigonometric functions are periodic for real arguments, e.g. the tangens has a period of $\pi$, so repetition of $y$ values due to the tangens might occur for $y_m$ if $$ \log(m) = \log(k) + n \pi \quad m \in \mathbb{N}, n\in \mathbb{Z} $$ which means $$ m = k \, (b^\pi)^n \in \mathbb{N} $$ where $b$ is the base of $\log$. I believe $b^\pi$ is not a rational number, so $(b^\pi)^n$ is not a natural number for $n \ne 0$. So I do not expect repetition due to the periodicity of the tangens.

This leaves open repetition due to periodicity of the sine or cosine function. Analyzing that seems complicated.

As your calculator uses finite representations of real numbers it seems more plausible to me that if repetition occurs, it occurs due to rounding errors.

So I see no reason for your specific number to show up repeatedly other than a property of your calculator or you did not evaluate the $y_k$ properly.