SOLUTION: 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)

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)
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
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)
***
let a=alpha
cos a=1/5 (in quadrant IV)
a≈281.54˚
sin a=-√(1-cos^2b)=-√(1-1/25)=-√(24/25)=-√24/5
..
sin b=1/4
b=14.48˚
cos b=√(1-sin^2b)=√(1-1/16)=√(15/16)=√15/4
..
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
calculator check:
sin(a-b)≈sin(281.54-14.48)≈sin(267.06)≈-0.9987
Exact value=-(√360+1)/20≈-0.9987
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