document.write( "Question 784570: how do you solve for x algebraically for : 2sin^2x+5sinx=3 \n" ); document.write( "

Algebra.Com's Answer #477288 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"2%28sin%28x%29%29%5E2%2B5sin%28x%29=3\"$
\n" ); document.write( "If you make a change of variable, it gets easier to write and easier to see the solution.
\n" ); document.write( "Substititing $\"y=sin%28x%29\"$ , the equation turns into
\n" ); document.write( " $\"2y%5E2%2B5y=3\"$
\n" ); document.write( "You probably know what to do with that equation.
\n" ); document.write( "I would do this:
\n" ); document.write( " $\"2y%5E2%2B5y=3\"$ --> $\"2y%5E2%2B5y-3=0\"$ --> $\"%282y-1%29%28y%2B3%29=0\"$
\n" ); document.write( "From that last, factored equation, we get two solutions in terms of $\"y\"$ :
\n" ); document.write( " $\"2y-1=0\"$ --> $\"2y=1\"$ --> $\"y=1%2F2\"$ and
\n" ); document.write( " $\"y%2B3=0\"$ --> $\"y=-3\"$
\n" ); document.write( "In terms of $\"x\"$ , $\"y=sin%28x%29=-3\"$ makes no sense.
\n" ); document.write( "On the other hand, from the other $\"y\"$ value we get many values for $\"x\"$
\n" ); document.write( " $\"y=sin%28x%29=1%2F2\"$
\n" ); document.write( "To begin with, if we just stick to the first turn, with $\"0%5Eo%3C=x%3C360%5Eo\"$ , we have 2 solutions: $\"highlight%2830%5Eo%29\"$ and $\"highlight%28150%5Eo%29\"$ , because
\n" ); document.write( " $\"sin%2830%5Eo%29=1%2F2\"$ and $\"sin%28150%5Eo%29=1%2F2\"$
\n" ); document.write( "If using radians rather than degrees, for $\"0%3C=x%3C2pi\"$ , the soluttions are $\"highlight%28pi%2F6%29\"$ and $\"highlight%285pi%2F6%29\"$ , because
\n" ); document.write( " $\"sin%28pi%2F6%29=1%2F2\"$ and $\"sin%285pi%2F6%29=1%2F2\"$
\n" ); document.write( "If you are not restricted to the first turn, there are infinite solutions, because co-terminal angles have the same trigonometric function values, so that all the solutions could be written as
\n" ); document.write( " $\"%284k%2B1%2990%5Eo+%2B-+60%5Eo\"$ or $\"%284k%2B1%29pi%2F2+%2B-+pi%2F3\"$ for any integer $\"k\"$ .
\n" ); document.write( "(Making $\"k=0\"$ gives you the solutions in the first turn). \n" ); document.write( "

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