document.write( "Question 1060046: Find all solutions of the following equation:\r
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document.write( "2 cos^2(x) - 3 cos(x) = 2 sin(x)cos(x) - 3 sin(x)\r
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document.write( "I am absolutely clueless as to how to solve this. Please help!! \n" );
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Algebra.Com's Answer #675108 by jim_thompson5910(35256)![]() ![]() ![]() You can put this solution on YOUR website! \n" ); document.write( "First get everything to one side\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "2 cos^2(x) - 3 cos(x) = 2 sin(x)cos(x) - 3 sin(x) \r \n" ); document.write( "\n" ); document.write( "2 cos^2(x) - 3 cos(x) - 2 sin(x)cos(x) + 3 sin(x) = 0\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Now separate the terms into two groups and factor by grouping\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "2 cos^2(x) - 3 cos(x) - 2 sin(x)cos(x) + 3 sin(x) = 0\r \n" ); document.write( "\n" ); document.write( "(2 cos^2(x) - 3 cos(x)) + (-2 sin(x)cos(x) + 3 sin(x)) = 0\r \n" ); document.write( "\n" ); document.write( "cos(x)(2 cos(x) - 3) + (-2 sin(x)cos(x) + 3 sin(x)) = 0\r \n" ); document.write( "\n" ); document.write( "cos(x)(2 cos(x) - 3) -sin(x)(2 cos(x) - 3) = 0\r \n" ); document.write( "\n" ); document.write( "(cos(x) - sin(x))(2 cos(x) - 3) = 0\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "After everything is factored, you can set each factor equal to 0 and solve for x\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Set the first factor equal to 0 \n" ); document.write( "cos(x) - sin(x) = 0 \n" ); document.write( "cos(x) = sin(x) \n" ); document.write( "Then use the unit circle to determine that the solutions occur at x = pi/4 and x = 5pi/4\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Set the other factor equal to 0 and solve for x \n" ); document.write( "2 cos(x) - 3 = 0 \n" ); document.write( "2 cos(x) = 3 \n" ); document.write( "cos(x) = 3/2 \n" ); document.write( "cos(x) = 1.5 ... no solutions \n" ); document.write( "Recall that the range of cosine has 1 as the largest value. So 1.5 is not possible. \r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "The only two solutions in the interval [0,2pi) are\r \n" ); document.write( "\n" ); document.write( "x = pi/4 \n" ); document.write( "x = 5pi/4\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Tack on \"+2pi*n\" to handle the rest of the solutions\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "So the entire solution set is \n" ); document.write( "x = pi/4 + 2pi*n \n" ); document.write( "x = 5pi/4 + 2pi*n \n" ); document.write( "where n is any integer \n" ); document.write( " \n" ); document.write( " |