document.write( "Question 625851: If tan α = 12/5, π < α < 3 π/2, and sin β = 2/3, 0< β < π/2, find\r
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Algebra.Com's Answer #393818 by Edwin McCravy(20060) $\"\"$ $\"About$

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document.write( "If tan α = 12/5, π < α < 3 π/2, and sin β = 2/3, 0< β < π/2, find\r\n" );
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document.write( "We draw the graphs of the angles,  is in the third\r\n" );
document.write( "quadrant and  is in the first:\r\n" );
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document.write( "Next we draw perpendiculars to the x-axis, which creates right\r\n" );
document.write( "triangles\r\n" );
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document.write( "Since the tangent is opposite/adjacent or y/x, we take the opposite side of \r\n" );
document.write( " to be -12 (negative because it goes downward from the x-axis), and\r\n" );
document.write( "the adjacent to be -5 (negative because it goes left from the origin).  \r\n" );
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document.write( "Since the sine is opposite/hypotenuse or y/r, we take the opposite side of \r\n" );
document.write( " to be +2 (positive because it goes upward from the x-axis), and\r\n" );
document.write( "the hypotenuse r to be 3 (the hypotenuse r is ALWAYS taken positive).\r\n" );
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document.write( "Next we use the Pythagorean theorem to calculate the hypotenuse r for \r\n" );
document.write( "and the adjacent side x for .\r\n" );
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document.write( "r² = x² + y²            r² = x² + y²\r\n" );
document.write( "r² = (-5)²+(-12)²       3² = x² + 2²\r\n" );
document.write( "r² = 25 + 144            9 = x² + 4  \r\n" );
document.write( "r² = 169                 5 = x² \r\n" );
document.write( " r = 13                   = x\r\n" );
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document.write( "Now we can answer the first two questions by merely looking at the triangles,\r\n" );
document.write( "and knowing sine = opposite/hypotenuse and cosine = adjacent/hypotenuse,\r\n" );
document.write( "using x for adjacent, y for opposite and r for hypotenuse:\r\n" );
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document.write( " A) sin α = opp/hyp = y/r = (-12)/13 = -12/13\r\n" );
document.write( " B) cos β = adj/hyp = x/r = /3 \r\n" );
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document.write( "To do the others we have to use identities and then we can\r\n" );
document.write( "substitute as above.  I will do C):\r\n" );
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document.write( " C) cos 2β = cos²β-sin²β = (/3)²-(2/3)² = 5/9-4/9=1/9\r\n" );
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document.write( "Now you can do the others by yourself, using the proper identities and\r\n" );
document.write( "the above graphs:\r\n" );
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document.write( " D) sin α/2\r\n" );
document.write( " E) cos (α+β)\r\n" );
document.write( " F) sin (α+β)/2 \r\n" );
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document.write( "Edwin

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