document.write( "Question 1019974: I need help on solving this trig equation:
\n" ); document.write( "(sin2x)(cosx)+(cos2x)(sinx)=(-1/2) \n" ); document.write( "

Algebra.Com's Answer #635912 by KMST(5328) $\"\"$ $\"About$

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The left side of that equation should remind you of the trigonometric identity
\n" ); document.write( " $\"sin%28A%2BB%29=sin%28A%29cos%28B%29%2Bcos%28A%29sin%28B%29\"$ .
\n" ); document.write( "Making $\"system%28A=2x%2CB=x%29\"$ (or the other way around),
\n" ); document.write( " $\"sin%282x%2Bx%29=sin%282x%29cos%28x%29%2Bcos%282x%29sin%28x%29\"$ , or
\n" ); document.write( " $\"sin%282x%29cos%28x%29%2Bcos%282x%29sin%28x%29=sin%283x%29\"$ .
\n" ); document.write( "So, we can re-write the equation as
\n" ); document.write( " $\"sin%283x%29=-1%2F2\"$ .
\n" ); document.write( "In the $\"%22%5B%22\"$ $\"-pi\"$ $\"%22+%2C%22\"$ $\"pi\"$ $\"%22%5D%22\"$ interval, the solutions for $\"3x\"$ are
\n" ); document.write( " $\"3x=-pi%2F6\"$ or $\"3x=-5pi%2F6\"$ .
\n" ); document.write( "We could write that as $\"3x=-pi%2F2+%2B-+pi%2F3\"$ .
\n" ); document.write( "Since sine has a period of $\"2pi\"$ ,
\n" ); document.write( "other values for $\"3x\"$ differing in any multiple of $\"2pi\"$ also have $\"sin%283x%29=-1%2F2\"$ .
\n" ); document.write( "All the solutions can be written as
\n" ); document.write( " $\"3x=-pi%2F2+%2B-pi%2F3+%2B2k%2Api=%284k-1%29pi%2F2+%2B-+pi%2F3\"$ for any integer $\"k\"$ .
\n" ); document.write( "So, $\"highlight%28x=%284k-1%29%2Api%2F2+%2B-+pi%2F9%29\"$ for any integer $\"k\"$ , or
\n" ); document.write( " $\"highlight%28x=%2812k-3+%2B-+2%29%2Api%2F18%29\"$ for any integer $\"k\"$ . \n" ); document.write( "

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