document.write( "Question 810211: Determine sin(2x)
\n" ); document.write( "when cos(x)=-1/3 and sin(x) is negative \n" ); document.write( "
Algebra.Com's Answer #487989 by DrBeeee(684)\"\" \"About 
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Use the double angle formula
\n" ); document.write( "(1) sin(2*x) = 2*sin(x)*cos(x)
\n" ); document.write( "and the givens
\n" ); document.write( "(2) cos(x) = -1/3 and
\n" ); document.write( "(3) sin(x) < 0
\n" ); document.write( "Use \"sin%5E2%28x%29+%2B+cos%5E2%28x%29+=+1\" to get
\n" ); document.write( "(4) \"sin%5E2%28x%29+%2B+%28-1%2F3%29%5E2+=+1\" or
\n" ); document.write( "(5) \"sin%5E2%28x%29++=+1+-1%2F9\" or
\n" ); document.write( "(6) \"sin%5E2%28x%29+=+8%2F9\" or
\n" ); document.write( "(7) \"sin%28x%29+=+-+sqrt%288%29%2F3\"
\n" ); document.write( "where we select the negative value in (7) because of the given condition (3).
\n" ); document.write( "Now put sin(x) and cos(x) into (1) to get
\n" ); document.write( "(8) sin(2*x) = 2*(- sqrt(8)/3)*(-1/3) or
\n" ); document.write( "(9)\"+sin%282%2Ax%29+=+2%2Asqrt%288%29%2F9\" or
\n" ); document.write( "(10) \"sin%282%2Ax%29+=+4%2Asqrt%282%29%2F9\"
\n" ); document.write( "Let's use numerical values to check the answer.
\n" ); document.write( "From (2) we get
\n" ); document.write( "(11) x = arccos(-1/3) or
\n" ); document.write( "(12) x = 109.47+
\n" ); document.write( "Then we get
\n" ); document.write( "(13) sin(x) = -0.9428
\n" ); document.write( "and
\n" ); document.write( "(14) sin(2x) = sin(218.94) or
\n" ); document.write( "(15) sin(2x) = 0.6285
\n" ); document.write( "Now
\n" ); document.write( "Is (sin(2*x) = 2*sin(x)*cos(x))?
\n" ); document.write( "Is (0.6285 = 2*(-09428)*(-1/3))?
\n" ); document.write( "Is (0.6285 = 0.6285)? Yes
\n" ); document.write( "Answer:\"sin%282%2Ax%29+=+4%2Asqrt%282%29%2F9\" \r
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