Question 1125838
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First equation....<br>
{{{sin(2x)^4-2*sin(2x)^2+1 = 0}}}
{{{(sin(2x)^2-1)^2 = 0}}}
{{{sin(2x)^2-1 = 0}}}
{{{(sin(2x)-1)(sin(2x)+1) = 0}}}
{{{sin(2x) = 1}}}  or  {{{sin(2x) = -1}}}<br>
0 < x < 2pi  -->  0 < 2x < 4pi<br>
On (0,4pi), sin(x) is 1 or -1 at pi/2, 3pi/2, 5pi/2, and 7pi/2.<br>
On (0,2pi), sin(2x) is 1 or -1 at pi/2, 3pi/2, 5pi/2, and 7pi/2.<br>
So the solution set for this equation, for x on (0,2pi), is<br>
{pi/4, 3pi/4, 5pi/4, 7pi/4}<br>
Second equation....<br>
{{{(3tan(3x))^2 = 27}}}
{{{9(tan(3x))^2 = 27}}}
{{{(tan(3x))^2 = 3}}}
{{{tan(3x) = sqrt(3)}}} or {{{tan(3x) = -sqrt(3)}}}<br>
0 < x < 2pi  -->  0 < 3x < 6pi<br>
On (0,6pi), tan(x) is sqrt(3) or -sqrt(3) at pi/3, 2pi/3, 4pi/3, 5pi/3, 7pi/3, 8pi/3, 10pi/3, 11pi/3, 13pi/3, 14pi/3, 16pi/3, and 17pi/3.<br>
On (0,2pi), tan(3x) is sqrt(3) or -sqrt(3) at pi/9, 2pi/9, 4pi/9, 5pi/9, 7pi/9, 8pi/9, 10pi/9, 11pi/9, 13pi/9, 14pi/9, 16pi/9, and 17pi/9.<br>
So the solution set for this equation, for x on (0,2pi), is<br>
{pi/9, 2pi/9, 4pi/9, 5pi/9, 7pi/9, 8pi/9, 10pi/9, 11pi/9, 13pi/9, 14pi/9, 16pi/9, 17pi/9}<br>