document.write( "Question 803728: if x is in the first and y in the third quadrant, sinx=3/5, and cosy=-5/13, find the exact value of sin(x+y), cos(x+y), and tan(x+y)?
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Algebra.Com's Answer #484466 by lwsshak3(11628) $\"\"$ $\"About$

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if x is in the first and y in the third quadrant, sinx=3/5, and cosy=-5/13, find the exact value of sin(x+y), cos(x+y), and tan(x+y)?
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\n" ); document.write( "sinx=3/5
\n" ); document.write( "so, you are working with a 3-4-5 reference right triangle in quadrant
\n" ); document.write( "cosx=4/5
\n" ); document.write( "tanx=sinx/cosx=3/4
\n" ); document.write( "..
\n" ); document.write( "cosy=-5/13
\n" ); document.write( "so, you are working with a 5-12-13 reference right triangle in quadrant III
\n" ); document.write( "siny=-12/13
\n" ); document.write( "tany=-12/-5=12/5
\n" ); document.write( "..
\n" ); document.write( "Identity: sin(x+y)=sinx*cosy+cosx*siny
\n" ); document.write( " $\"%283%2F5%29%28-5%2F13%29%2B%284%2F5%29%28-12%2F13%29\"$
\n" ); document.write( " $\"%28-15%2F65%29%2B%28-48%2F65%29=-63%2F65\"$
\n" ); document.write( "..
\n" ); document.write( "Identity: cos(x+y)=cosx*cosy-sinx*siny
\n" ); document.write( " $\"%284%2F5%29%28-5%2F13%29-%283%2F5%29%28-12%2F13%29\"$
\n" ); document.write( " $\"%28-20%2F65%29-%28-36%2F65%29=16%2F65\"$
\n" ); document.write( "..
\n" ); document.write( "Identity: tan(x+y)= $\"%28tanx%2Btany%29%2F%281-tanx%2Atany%29\"$
\n" ); document.write( " $\"%28%283%2F4%29%2B%2812%2F5%29%29%2F%281-%283%2F4%29%2A%2812%2F5%29%29\"$
\n" ); document.write( " $\"%28%2815%2F20%29%2B%2848%2F20%29%29%2F%281-%2836%2F20%29%29\"$
\n" ); document.write( " $\"%2863%2F20%29%2F%28-16%2F20%29=-63%2F16\"$
\n" ); document.write( "..
\n" ); document.write( "Calculator check:
\n" ); document.write( "sinx=3/5
\n" ); document.write( "x≈36.87˚
\n" ); document.write( "cosy=-5/13
\n" ); document.write( "y≈67.38˚+180≈247.38˚
\n" ); document.write( "x+y≈284.25˚
\n" ); document.write( "..
\n" ); document.write( "sin(x+y)≈sin(284.25˚)≈-0.9692..
\n" ); document.write( "Exact value as calculated=-63/65≈-0.9692..
\n" ); document.write( "..
\n" ); document.write( "cos(x+y)≈cos(284.25)≈0.2461..
\n" ); document.write( "Exact value as calculated=16/65≈0.2461..
\n" ); document.write( "..
\n" ); document.write( "tan(x+y)≈tan(284.25)≈-3.9375..
\n" ); document.write( "Exact value as calculated=-63/16≈-3.9375.. \n" ); document.write( "

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