Question

Use the given information to find the exact value of the expression. sin α = 2129 21 29 , α lies in quadrant II, and cos β = 1517 15 17 , β lies in quadrant I Find sin (α - β).

Answer 1

The correct question is
Use the given information to find the exact value of the expression. sin α = 21/29, α lies in quadrant II, and cos β = 15/17, β lies in quadrant I Find sin (α - β).

we know that
sin(α − β) = sin α cos β − cos α sin β

α lies in quadrant II
so
cos α  is negative
sin α  is positive

β lies in quadrant I
so
cos β  is positive
sin β   is positive

step 1
find sin β
cos β=15/17
sin² β+cos² β=1-----------> sin² β=1-cos² β----> sin² β=1-(15/17)²
sin² β=1-225/289-----> 64/289
sin β=8/17

step 2
find cos α
sin α = 21/29
cos² α + sin² α=1----> cos² α=1-sin² α---> cos² α=1-(21/29)²---> 1-441/841
cos² α=400/841------> cos α=-20/29  (remember cos α is negative)

step 3
find sin(α − β) 
sin α = 21/29   cos α=-20/29
sin β=8/17       cos β=15/17

sin(α − β) = [21/29]*[15/17] − [-20/29*]*[8/17]
sin(α − β) = [315/493] − [-160/493]
sin(α − β) = 475/493

the answer is
sin(α − β) = 475/493