document.write( "Question 639149: Solve:
\n" ); document.write( "sin 3x*cos x + cos 3x*sin x=-1/2 \n" ); document.write( "

Algebra.Com's Answer #402598 by jsmallt9(3758) $\"\"$ $\"About$

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The easiest way to solove this is by recognizing that the left side of the equation fits the pattern for sin(A+B):
\n" ); document.write( "sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
\n" ); document.write( "with the \"A\" being \"3x\" and the \"B\" being \"x\". So by sin((A+B) the left side is equal to:
\n" ); document.write( "sin(3x+x) = -1/2
\n" ); document.write( "or simply
\n" ); document.write( "sin(4x) = -1/2

\n" ); document.write( "This equation we can solve. We should recognize that 1/2 is a special angle value for sin and that the reference angle will be $\"pi%2F6\"$ . Since the sin is negative, we know that the angle terminates in the 3rd or 4th quadrants. Putting the reference angle and the quadrants together we get:
\n" ); document.write( " $\"4x+=+pi+%2B+pi%2F6+%2B+2pi%2An\"$ (for the 3rd quadrant)
\n" ); document.write( "which simplifies to
\n" ); document.write( " $\"4x+=+7pi%2F6+%2B+2pi%2An\"$ (for the 3rd quadrant)
\n" ); document.write( "and
\n" ); document.write( " $\"4x+=+-pi%2F6+%2B+2pi%2An\"$ (for the 4th quadrant)
\n" ); document.write( "(Instead of $\"-pi%2F6\"$ we could also have used $\"2pi+-+pi%2F6\"$ or $\"11pi%2F6\"$ )

\n" ); document.write( "Now we just divide both sides by 4 (or multiply by 1/4):
\n" ); document.write( " $\"x+=+7pi%2F24+%2B+pi%2An%2F2\"$
\n" ); document.write( "and
\n" ); document.write( " $\"x+=+-pi%2F24+%2B+pi%2An%2F2\"$ \n" ); document.write( "

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