document.write( "Question 773762: Find all solutions to the equation in the interval (0,2pi).
\n" ); document.write( "sin(x+(pi/6))-sin(x-(pi/6))=1/2
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #471875 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
If we apply the trigonometric identities
\n" ); document.write( " $\"sin%28a%2Bb%29+=+sin%28a%29cos%28b%29+%2B+cos%28a%29sin%28b%29\"$
\n" ); document.write( " $\"sin%28a-b%29+=+sin%28a%29cos%28b%29+-+cos%28a%29sin%28b%29\"$
\n" ); document.write( "we get
\n" ); document.write( "

\n" ); document.write( "

\n" ); document.write( "With that, $\"sin%28x%2Bpi%2F6%29-sin%28x-pi%2F6%29=1%2F2\"$ turns into
\n" ); document.write( "

\n" ); document.write( "

\n" ); document.write( " $\"%281%2F2%29%2Acos%28x%29%2B%281%2F2%29%2Acos%28x%29=1%2F2\"$
\n" ); document.write( " $\"cos%28x%29=1%2F2%29\"$
\n" ); document.write( "We know that $\"cos%28pi%2F3%29=1%2F2\"$ , so $\"highlight%28x=pi%2F3%29\"$ in quadrant I is a solution.
\n" ); document.write( "We know that there is an angle with the same cosine in quadrant IV.
\n" ); document.write( "I think of that angle as $\"x=-pi%2F3\"$ , but the coterminal angle between $\"0\"$ and $\"2pi\"$ is
\n" ); document.write( " $\"x=2pi-pi%2F3\"$ --> $\"highlight%28x=5pi%2F3%29\"$
\n" ); document.write( "That is the other solution.
\n" ); document.write( "
\n" ); document.write( "Verification:
\n" ); document.write( " $\"x=pi%2F3\"$ --> $\"system%28x%2Bpi%2F6=pi%2F3%2Bpi%2F6%2Cx-pi%2F6=pi%2F3-pi%2F6%29\"$ --> $\"system%28x%2Bpi%2F6=pi%2F2%2Cx-pi%2F6=pi%2F6%29\"$ --> $\"system%28sin%28x%2Bpi%2F6%29=1%2Csin%28x-pi%2F6%29=1%2F2%29\"$ --> $\"sin%28x%2Bpi%2F6%29-sin%28x-pi%2F6%29=1-1%2F2=1%2F2\"$
\n" ); document.write( "
\n" ); document.write( " $\"x=5pi%2F3\"$ --> $\"system%28x%2Bpi%2F6=5pi%2F3%2Bpi%2F6%2Cx-pi%2F6=5pi%2F3-pi%2F6%29\"$ --> $\"system%28x%2Bpi%2F6=11pi%2F6%2Cx-pi%2F6=3pi%2F2%29\"$ --> $\"system%28sin%28x%2Bpi%2F6%29=-1%2F2%2Csin%28x-pi%2F6%29=-1%29\"$ --> $\"sin%28x%2Bpi%2F6%29-sin%28x-pi%2F6%29=-1%2F2-%28-1%29=-1%2F2%2B1=1%2F2\"$ \n" ); document.write( "

\n" );