document.write( "Question 733572: If tan α = −5/12 and cot β = 8/15
\n" ); document.write( " for a second-quadrant angle α and a third-quadrant angle β, find sin(α + β) \n" ); document.write( "
Algebra.Com's Answer #448481 by lwsshak3(11628)\"\" \"About 
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If tan α = −5/12 and cot β = 8/15
\n" ); document.write( "for a second-quadrant angle α and a third-quadrant angle β, find sin(α + β)
\n" ); document.write( "***
\n" ); document.write( "Identity: sin(α + β)=sin α cos β+cos α sin β
\n" ); document.write( "let O=opposite side
\n" ); document.write( "let A=adjacent side
\n" ); document.write( "let H=hypotenuse
\n" ); document.write( "..
\n" ); document.write( "tan α = −5/12 (in Q2)=O/-A
\n" ); document.write( "O=5, A=-12
\n" ); document.write( "H=√(O^2+A^2)=√(25+144)=√169=13
\n" ); document.write( "sin α=O/H=5/13
\n" ); document.write( "cos α=A/H=-12/13
\n" ); document.write( "..
\n" ); document.write( "cot β = 8/15(in Q3)=-A/-O
\n" ); document.write( "A=-8, O=-15
\n" ); document.write( "H=√(O^2+A^2)=√(225+64)=√289=17
\n" ); document.write( "sin β=O/H=-15/17
\n" ); document.write( "cos β=A/H=-8/17
\n" ); document.write( "..
\n" ); document.write( "sin(α + β)=sin α cos β+cos α sin β
\n" ); document.write( "=(5/13*-8/17)+(-12/13*-15/17)
\n" ); document.write( "=-40/221+180/221
\n" ); document.write( "sin(α + β)=140/221
\n" ); document.write( "..
\n" ); document.write( "Check with calculator:
\n" ); document.write( "tan α=-5/12 (in Q2)
\n" ); document.write( "α≈157.38º
\n" ); document.write( "cot β = 8/15(in Q3)
\n" ); document.write( "β≈241.93º
\n" ); document.write( "α+β≈157.38+241.93≈399.31
\n" ); document.write( "reference angle in Q1≈39.31º
\n" ); document.write( "sin 39.31=0.6335
\n" ); document.write( "sin(α + β)=140/221≈0.6335 \n" ); document.write( "
\n" );