What's their relation ?

Algebra Level 3

Over all real numbers $x$ , what is the relation between $\sin( \cos(x))$ and $\cos(\sin(x))$ ?

cos(sin(x)) <= sin(cos(x)) cos(sin(x)) < sin(cos(x)) cos(sin(x)) > sin(cos(x)) cos(sin(x)) >= sin(cos(x))

1 solution

Sheikh Asif Imran Shouborno
May 19, 2014

We can have multiple procedures, but it seems that most of us are adsorbent to this way:

As we know, RANGE of $\sin x$ and $\cos x$ functions are $[-1,1]$ , which will now be our RESTRICTED DOMAIN for the latter imposed $\sin$ & $\cos$ functions.

Simply, we derive, where $\sin(\cos(x))$ function will range some value around $0$ , $\cos(\sin(x))$ function will range around and similarly close to $1$ .

So, there remains $0$ probability of the functions being equal, and we come up the decision that, $\sin(\cos(x))<\cos(\sin(x))$ .

That's all!

What's their relation ?

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