document.write( "Question 817936: How do I find sin (x-y) and tan (x+y) if cos x=(-1/3) and tan y= (1/2) with x being a quadrant II angle and y a quadrant III angle? Thank you for your help! \n" ); document.write( "

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How do I find sin (x-y) and tan (x+y) if cos x=(-1/3) and tan y= (1/2) with x being a quadrant II angle and y a quadrant III angle?
\n" ); document.write( "***
\n" ); document.write( "Identity: sin(x-y)=sinxcosy-cosxsiny
\n" ); document.write( "Identity: tan(x+y)=(tanx+tany)/(1-tanxtany)
\n" ); document.write( "..
\n" ); document.write( "cosx=(-1/3)(In quadrant II where cos<0, sin>0, tan<0)
\n" ); document.write( "sinx=√(1-cos^2x)=√(1-1/9)=√(8/9)=√8/3
\n" ); document.write( "tanx=sinx/cosx=√8/-1=-√8
\n" ); document.write( "..
\n" ); document.write( "tany=1/2(In quadrant III where sin<0, cos<0, tan>0)
\n" ); document.write( "hypotenuse of reference right triangle in quadrant III=√((1^2)+(2^2))=√(1+4)=√5
\n" ); document.write( "siny=-1/√5=-√5/5
\n" ); document.write( "cosy=-2/√5=-2√5/5
\n" ); document.write( "..
\n" ); document.write( "sin(x-y)=√8/3*-2√5/5-(-1/3)*-√5/5=-(2√40+√5)/15
\n" ); document.write( "tan(x+y)=(-√8+1/2)/(1-(-√8)*1/2=(-√8+1/2)/(1+√8/2)=((-2√8+1)/2)/((2+√8)/2)
\n" ); document.write( "=(-2√8+1)/(2+√8)
\n" ); document.write( "..
\n" ); document.write( "Calculator check:
\n" ); document.write( "cosx=-1/3
\n" ); document.write( "x≈109.47˚(Q2)
\n" ); document.write( "tany=1/2
\n" ); document.write( "y≈206.57˚(Q3)
\n" ); document.write( "x+y≈316.04˚
\n" ); document.write( "x-y≈-97.1˚
\n" ); document.write( "..
\n" ); document.write( "sin(x-y)≈sin(-97.1)≈-0.9923..
\n" ); document.write( "exact value as calculated=-(2√40+√5)/15≈-0.9923..
\n" ); document.write( "..
\n" ); document.write( "tan(x+y)≈tan(316.04)≈-0.964..
\n" ); document.write( "exact value as calculated=(-2√8+1)/(2+√8)≈-0.964..
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