document.write( "Question 868732: Solve cos(2x)+sin(x)=1in[0,2pi) \n" ); document.write( "

Algebra.Com's Answer #523827 by KMST(5328) $\"\"$ $\"About$

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Use the trigonometric identity $\"cos%282x%29=%28cos%28x%29%29%5E2-%28sin%28x%29%29%5E2\"$ .
\n" ); document.write( "Since $\"%28cos%28x%29%29%5E2=1-%28sin%28x%29%29%5E2\"$ , it can be re-arranged to
\n" ); document.write( " $\"cos%282x%29=1-2%28sin%28x%29%29%5E2\"$ .
\n" ); document.write( "Substituting into $\"+cos%282x%29%2Bsin%28x%29=1\"$ you get
\n" ); document.write( " $\"1-2%28sin%28x%29%29%5E2%2Bsin%28x%29=1\"$ <---> $\"-2%28sin%28x%29%29%5E2%2Bsin%28x%29=0\"$ (subtracting $\"1\"$ from both sides of the equal sign)
\n" ); document.write( "That is a quadratic equation in $\"sin%28x%29\"$ .
\n" ); document.write( "It looks complicated, but if you rename $\"y=sin%28x%29\"$ ,
\n" ); document.write( "you can re-write it and it looks very simple:
\n" ); document.write( " $\"-2y%5E2%2By=0\"$ <--> $\"2y%5E2-y-0\"$ <--> $\"y%282y-1%29=0\"$ --> $\"system%28y=0%2C%22or%22%2Cy=1%2F2%29\"$
\n" ); document.write( "So, $\"-2%28sin%28x%29%29%5E2%2Bsin%28x%29=0\"$ is true when
\n" ); document.write( " $\"system%28sin%28x%29=0%2C%22or%22%2Csin%28x%29=1%2F2%29\"$ .
\n" ); document.write( "In $\"%22%5B+0+%2C%22\"$ $\"2pi\"$ $\"%22%29%22\"$ ,
\n" ); document.write( " $\"sin%28x%29=0\"$ --> $\"system%28highlight%28x=0%29%2C%22or%22%2Chighlight%28x=pi%29%29\"$ and
\n" ); document.write( " $\"sin%28x%29=1%2F2\"$ --> $\"system%28highlight%28x=pi%2F6%29%2C%22or%22%2Chighlight%28x=5pi%2F6%29%29\"$ \n" ); document.write( "

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