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So in trig, say I have an acute angle $X$. And one can intuitively conclude that $\sin X + \cos X \ge 1$ but how does the fact that $(\sin X + \cos X)^2 = 1 + 2 \sin X \cos X$ tell me that it is true that $\sin X + \cos X \ge 1$? I don't quite see the connection.

Thanks

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If $x$ is acute $\sin x$ and $\cos x$ are both positive, and therefor $$2\sin x\cos x\ge 0\\ 1+2\sin x\cos x\ge 1\\ \sin^2x+\cos^2x+2\sin x\cos x\ge 1\\ \left(\sin x+\cos x\right)^2\ge 1$$ Where both $\sin x$ and $\cos x$ are positive, so is their addition: $$\sin x+\cos x\ge 1$$

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  • $\begingroup$ But just because sinX and cosX are greater than 0 doesn't guarantee that their addition is greater than 0? $\endgroup$ – Kat Oct 13 '13 at 3:50
  • $\begingroup$ @Kat Order is preserved in addition so if $a>0$ and $b>0$ then $a+b>0+0=0$. Now if $c=a+b$ is positive, and $c^2\ge1^2=1$, then $c\ge1$. (If $c$ is negative, then $c\le-1$.) $\endgroup$ – Carlos Eugenio Thompson Pinzón Oct 13 '13 at 4:01
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For an acute angle $X$, $\sin X \ge 0$ and $\cos X \ge 0$. So $1 + 2 \sin X \cos X \ge 1$...

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