Problem 11:
Solve the inequality\[2\cos x \le |\sqrt {1+\sin 2x} -\sqrt {1- \sin 2x}|\]
BdMO National Higher Secondary 2007/11
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nafistiham
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Re: BdMO National Higher Secondary 2007/11
I did it like this.
\[2cosx\leq|\sqrt{1+sin2x}-\sqrt{1-sin2x}|\]
by squaring and simplifying
\[ 2cos^2x\le 1-cos2x\]
\[\Rightarrow 2cos^2x\le2-2cos^2x\]
\[\Rightarrow cosx\le\frac{1}{\sqrt2}\]
edited.courtesy:bristy1588
\[2cosx\leq|\sqrt{1+sin2x}-\sqrt{1-sin2x}|\]
by squaring and simplifying
\[ 2cos^2x\le 1-cos2x\]
\[\Rightarrow 2cos^2x\le2-2cos^2x\]
\[\Rightarrow cosx\le\frac{1}{\sqrt2}\]
edited.courtesy:bristy1588
Last edited by nafistiham on Wed Jan 11, 2012 5:10 pm, edited 1 time in total.
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
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bristy1588
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Re: BdMO National Higher Secondary 2007/11
Tiham, I did not understand how u got from line 1 to line 2.nafistiham wrote:I did it like this.
\[2cosx\leq|\sqrt{1+sin2x}-\sqrt{1-sin2x}|\]
by squaring and simplifying
1. \[ 2cos^2x\le 1-cos2x\]
2. \[\Rightarrow 2cos^2x\le2-2cosx\]
3.\[\Rightarrow (cosx+1)(2cosx-1)\le0\]
is there any bug
$cos2x=2cos^2x-1$
Put this in your equation and check..
Bristy Sikder