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Given angle is in radians, how can we find intervals on which all such inequalities hold true?

$\alpha > \sin(\alpha)$, $\alpha < \sin(\alpha)$, $\alpha \leqq \sin(\alpha)$, $\alpha \geqq \sin(\alpha)$

And all the values where $\alpha = \sin(\alpha)$

Plus, same for $\cos, \tan$.

P.S. I understand that $\alpha \leqq 1$ for sine and cosine. But for $\tan$ and $\alpha < 1$ for sine and cosine I don't have a clue.

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    $\begingroup$ For $x>0$, $x>\sin x$ and for $x<0$, $x < \sin x$. $\endgroup$ – Dr Zafar Ahmed DSc Oct 6 at 10:42
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By the Leibniz test, for $0<x<2$, $$ x-\frac{x^3}6\le\sin x\le x-\frac{x^3}6+\frac{x^5}{120}<x $$ and for $x\ge 2>1\ge \sin x$ is trivially true. Similar holds for the negative half axis.

Thus your question about the sine has a very simple answer.


For the cosine and tangent these equations are transcendental, thus have no nice solutions. The same goes for any modification of the sine question, for instance asking when $α\geqq 2\sin α$

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