document.write( "Question 42653: Find the exact value of sin 2θ if cosθ = -(√5)/3 and 180° < θ < 270°.\r
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Algebra.Com's Answer #27763 by AnlytcPhil(1807)\"\" \"About 
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Find the exact value of sin 2θ if\r\n" );
document.write( "cosθ = -(√5)/3 and 180° < θ < 270°. \r\n" );
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document.write( "We need to use the formula\r\n" );
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document.write( "sin(2θ) = 2sinθ·cosθ\r\n" );
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document.write( "We know cosθ but not sinθ.  We have to find sinθ.\r\n" );
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document.write( "We are given 180° < θ < 270° so we know θ is in \r\n" );
document.write( "quadrant III.  So we draw the picture of θ in the\r\n" );
document.write( "third quadrant\r\n" );
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document.write( "    _ |\r\n" );
document.write( "  -√5 |\r\n" );
document.write( "-------------\r\n" );
document.write( "  |  /|\r\n" );
document.write( "  | /3|\r\n" );
document.write( "  |/  |\r\n" );
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document.write( "Since the cosine is x/r or adjacent/hypotenuse, we\r\n" );
document.write( "                       _             _ \r\n" );
document.write( "put the numerator of -√5/3, namely -√5 the on the x-side\r\n" );
document.write( "(or adjacent side) and 3 on the r, the radius vector (or\r\n" );
document.write( "hypotenuse).\r\n" );
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document.write( "We use the Pythagorean theorem to find the y-side (or the\r\n" );
document.write( "opposite side).\r\n" );
document.write( "      _______     ____________      ___     _\r\n" );
document.write( "y = ±√r² - x² = ±√(3)² - (-√5)² = ±√9-5 = ±√4 = ±2\r\n" );
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document.write( "We know to take the negative sign since y goes down\r\n" );
document.write( "from the x-axis, so we have\r\n" );
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document.write( "    _ |\r\n" );
document.write( "  -√5 |\r\n" );
document.write( "-------------\r\n" );
document.write( "  |  /|\r\n" );
document.write( "-2| /3|\r\n" );
document.write( "  |/  |\r\n" );
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document.write( "Now we know sinθ = -2/3\r\n" );
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document.write( "So\r\n" );
document.write( "sin(2θ) = 2sinθ·cosθ \r\n" );
document.write( "                     _ \r\n" );
document.write( "sin(2θ) = 2(-2/3)·(-√5/3)\r\n" );
document.write( "                     _ \r\n" );
document.write( "sin(2θ) = 2(-2/3)·(-√5/3)\r\n" );
document.write( "            _ \r\n" );
document.write( "sin(2θ) = 4√5/9\r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com
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