document.write( "Question 921705: find the exact value of sin(alpha-beta) given that cos(alpha)=1/5, a is between (-pi/2,0); sin beta=1/4, beta is between (0,pi/2) \n" ); document.write( "
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find the exact value of sin(alpha-beta) given that cos(alpha)=1/5, a is between (-pi/2,0); sin beta=1/4, beta is between (0,pi/2)
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\n" ); document.write( "let a=alpha
\n" ); document.write( "cos a=1/5 (in quadrant IV)
\n" ); document.write( "a≈281.54˚
\n" ); document.write( "sin a=-√(1-cos^2b)=-√(1-1/25)=-√(24/25)=-√24/5
\n" ); document.write( "..
\n" ); document.write( "sin b=1/4
\n" ); document.write( "b=14.48˚
\n" ); document.write( "cos b=√(1-sin^2b)=√(1-1/16)=√(15/16)=√15/4
\n" ); document.write( "..
\n" ); document.write( "sin(a-b)=(sin a*cos b)-(cos a*sin b)=(-√24/5*√15/4)-(1/5*1/4)=-(√360/20+1/20)=-(√360+1)/20
\n" ); document.write( "calculator check:
\n" ); document.write( "sin(a-b)≈sin(281.54-14.48)≈sin(267.06)≈-0.9987
\n" ); document.write( "Exact value=-(√360+1)/20≈-0.9987 \n" ); document.write( "
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