First, notice that $(sin x +cos x)^2=sin^2 x+cos^2 x+sin 2x=1+ frac 23=frac53$ .
Now, from what was given we have $sin x=frac13cos x$ và $cos x=frac13sin x$ .
Next, $(sin^2 x+cos^2 x)^3=1=sin^6 x+cos^6 x+3sin^2 x cos x+3cos^2 x sin x$ .
Now we substitute what we found above from the given:
$sin^6 x+cos^6+sin x +cos x=1$
$sin^6 x+cos^6=1-(sin x +cos x)$
$sin^6 x+cos^6=1-sqrt frac 53$
Not only is this not positive, but this is not even a rational number. What did I vì wrong? Thanks.
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asked Jun 20, 2013 at 19:31
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$(sin^2 x + cos^2 x)^3=sin^6 x + cos^6 x + 3sin^2 x cos^2 x$
answered Jun 20, 2013 at 19:36
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Should be $(sin^2 x+cos^2 x)^3=1=sin^6 x+cos^6 x+3sin^4 x cos^2 x+3cos^4 x sin^2 x$
answered Jun 20, 2013 at 19:35
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$sin^6x + cos^6x = (sin^2x)^3 + (cos^2x)^3 =(sin^2x + cos^2x)(sin^4x + cos^4x -sin^2xcos^2x)$
$sin^4x+cos^4x -sin^2xcos^2x = (sin^2x + cos^2x)^2 - 2sin^2xcos^2x -sin^2xcos^2x$
or $1-3sin^2xcos^2x = 1-3left(dfrac13 ight)^2 = dfrac23$.
edited Oct 2, 2013 at 17:43
answered Jul 21, 2013 at 8:31
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$egingroup$ It has to lớn be $1 - 1 / 3$, not $1 - (1 / 3) ^ 2$, the answer is $2 / 3$. $endgroup$
Oct 2, 2013 at 10:54
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