document.write( "Question 973867: Please help!!!!
\n" ); document.write( "What are all the solutions of the equation cos(2x)+1=sin(2x) in interval [0,2pi)? \n" ); document.write( "

Algebra.Com's Answer #595934 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"Let+me+write+%7B%7B%7Bu=2x\"$ for short, that way, I can simply write
\n" ); document.write( " $\"cos%28u%29%2B1=sin%28u%29\"$
\n" ); document.write( "Since $\"cos%28u%29%3E=-1\"$ , $\"sin%28u%29=cos%28u%29%2B1%3E=-1%2B1=0\"$ .
\n" ); document.write( "Also, since $\"sin%28u%29%3C=1\"$ , $\"sin%28u%29=cos%28u%29%2B1%3C=1\"$ ---> $\"cos%28u%29%3C=1-1=0\"$ .
\n" ); document.write( "With a negative cosine and a positive sine, $\"u\"$ must be in quadrant II.\r
\n" ); document.write( "\n" ); document.write( "Maybe $\"pi%2F2%3C=u%3C=pi\"$ as in

or maybe even $\"pi%2F2%2B2pi=5pi%2F2%3C=u%3C=3pi\"$ as in

\n" ); document.write( "Knowing that $\"sin%28u%29%3E=0\"$ ,
\n" ); document.write( " $\"cos%28u%29%2B1=sin%28u%29\"$ ---> $\"cos%28u%29%2B1=sqrt%281-cos%5E2%28u%29%29\"$ ---> $\"%28cos%28u%29%2B1%29%5E2=1-cos%5E2%28u%29\"$ ---> $\"cos%28u%29%5E2%2B2cos%28u%29%2B1=1-cos%5E2%28u%29\"$ ---> $\"cos%28u%29%5E2%2B2cos%28u%29=-cos%5E2%28u%29\"$ ---> $\"2cos%28u%29%5E2%2B2cos%28u%29=0\"$ ---> $\"2cos%28u%29%28cos%28u%29%2B1%29=0\"$ .
\n" ); document.write( "So, either $\"cos%28u%29=0\"$ , or $\"cos%28u%29%2B1=0\"$ <---> $\"cos%28u%29=-1\"$ .
\n" ); document.write( "
\n" ); document.write( "For quadrant II, $\"cos%28u%29=0\"$ ---> $\"system%28u=pi%2F2%2C%22or%22%2Cu=5pi%2F2%29\"$ , and
\n" ); document.write( " $\"u=2x=pi%2F2\"$ means $\"highlight%28x=pi%2F4%29\"$ , while
\n" ); document.write( " $\"u=2x=5pi%2F2\"$ means $\"highlight%28x=5pi%2F4%29\"$ .
\n" ); document.write( "
\n" ); document.write( "For quadrant II, $\"cos%28u%29=-1\"$ ---> $\"system%28u=pi%2C%22or%22%2Cu=3pi%29\"$ , and
\n" ); document.write( " $\"u=2x=pi\"$ means $\"highlight%28x=pi%2F2%29\"$ , while
\n" ); document.write( " $\"u=2x=3pi\"$ means $\"highlight%28x=3pi%2F2%29\"$ .
\n" ); document.write( "
\n" ); document.write( "If we think that $\"sin%282x%29\"$ and $\"cos%282x%29\"$ have a period of $\"pi\"$ ,
\n" ); document.write( "or if we graph $\"red%28y=cos%282x%29%2B1%29\"$ and $\"green%28y=sin%282x%29%29\"$ we see that it makes a lot of sense:
\n" ); document.write( " $\"graph%28700%2C300%2C-0.3%2C6.7%2C-1.1%2C2.2%2Ccos%282x%29%2B1%2Csin%282x%29%29\"$ \r
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