document.write( "Question 683200: solve -2sin^2x=3sinx+1 for exact solutions over the interval [0,2π]
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #423488 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"-2sin%5E2%28x%29=3sin%28x%29%2B1\"$
\n" ); document.write( "\"exact solutions\" is code for: This problem involves special angles and we should put away our calculators.

\n" ); document.write( "First we want to transform the equation into one or more equations of the form:
\n" ); document.write( "TrigFunction(expression) = number
\n" ); document.write( "This equation is in quadratic form for sin(x). So we'll start by getting one side to be zero. Adding $\"2sin%5E2%28x%29\"$ to each side:
\n" ); document.write( " $\"0+=+2sin%5E2%28x%29%2B3sin%28x%29%2B1+\"$
\n" ); document.write( "Now we factor. This factors fairly easily. If you have trouble seeing this then use a temporary variable:
\n" ); document.write( "Let q = sin(x). Then the equation becomes:
\n" ); document.write( " $\"0+=+2q%5E2%2B3q%2B1\"$
\n" ); document.write( "After you factor it replace the q's with sin(x)'s and you'll get:

\n" ); document.write( " $\"0+=+%282sin%28x%29%2B1%29%28sin%28x%29%2B1%29+\"$
\n" ); document.write( "From the Zero Product Property:
\n" ); document.write( "2sin(x) + 1 = 0 or sin(x) + 1 = 0
\n" ); document.write( "Solving these for sin(x) we get:
\n" ); document.write( "sin(x) = -1/2 or sin(x) = -1
\n" ); document.write( "These equations are in the desired form.

\n" ); document.write( "Next we find the general solution. As anticipated we have special angle values for sin in both equations. For sin(x) = -1/2 we should recognize the reference angle of $\"pi%2F6\"$ has a sin of 1/2. And since sin is negative in the 3rd and 4th quadrants our general solution for this equation is:
\n" ); document.write( " $\"x+=+pi%2Bpi%2F6%2B2pi%2An\"$ for the 3rd quadrant angles
\n" ); document.write( " $\"x+=+-pi%2F6%2B2pi%2An\"$ (or $\"2pi-pi%2F6%2B2pi%2An\"$ ) for the 4th quadrant angles
\n" ); document.write( "These simplify to:
\n" ); document.write( " $\"x+=+7pi%2F6%2B2pi%2An\"$ for the 3rd quadrant angles
\n" ); document.write( " $\"x+=+-pi%2F6%2B2pi%2An\"$ (or $\"11pi%2F6%2B2pi%2An\"$ ) for the 4th quadrant angles

\n" ); document.write( "For the equation sin(x) = -1 we should know that only $\"3pi%2F2\"$ (and co-terminal angles) will have a sin of -1. So the general solution for this is:
\n" ); document.write( " $\"x+=+3pi%2F2%2B2pi%2An\"$
\n" ); document.write( "These three general solution equations express the infinite set of angles that fit your equation.

\n" ); document.write( "Your problem asks for solutions over the interval [0,2π]. For this we use the general solution equations and replace the n's with integers until we find all the x's in the given interval.
\n" ); document.write( "For the equation $\"x+=+7pi%2F6%2B2pi%2An\"$ :
\n" ); document.write( "If n = 0 then x = $\"7pi%2F6\"$
\n" ); document.write( "If n = 1 (or other positive integers, x is greater than $\"2pi\"$
\n" ); document.write( "If n = -1 (or other negative integers, x is below 0
\n" ); document.write( "For the equation $\"x+=+-pi%2F6%2B2pi%2An\"$
\n" ); document.write( "If n = 0 (or any negative integer, x is below 0
\n" ); document.write( "If n = 1 then x = $\"11pi%2F6\"$
\n" ); document.write( "If n = 2 (or larger positive integers, x is greater than $\"2pi\"$
\n" ); document.write( "For the equation $\"x+=+3pi%2F2%2B2pi%2An\"$
\n" ); document.write( "If n = 0 then x is below $\"3pi%2F2\"$
\n" ); document.write( "If n = 1 (or other positive integers, x is greater than $\"2pi\"$
\n" ); document.write( "If n = -1 (or other negative integers, x is below 0
\n" ); document.write( "So there are only three solutions in the interval [0,2π]: $\"7pi%2F6\"$ , $\"11pi%2F6\"$ and $\"3pi%2F2\"$ \n" ); document.write( "

\n" );