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There is something I don't understand with arccos and inequalities.

Suppose I have this inequality

$cos(x) ≤ \frac{1}{2}$

Having $x = 90$, satisfies this since $cos(90) = 0$.

Then since arccos is defined on [-1,1], I should be able to arccos both sides, which gives me

$x ≤ 60$

But then when $x = 90$, the inequality isn't satisfied.

Why does this happen, even though I kept everything in degrees format?

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1 Answer 1

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The reason is that $\arccos$ is a decreasing function. So after you apply it to both sides, you need to switch the direction of the inequality. In general, for any decreasing function $f$, if $a \leq b$, then $f(a) \geq f(b)$. Examples of such $f$, besides $\arccos$ are: multiply by negative one [i.e. $m(x) = -x$] and take the recripical (i.e. $r(x) = 1/x$).

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