Find the exact value of cosu v given that sin u 35 with u

Find the exact value of cos(u + v) given that sin u = -3/5, with u = -3/5, with u in quadrant III, and sin v = -12/13, with in the quadrant IV. cos(u + v) =

Solution

given sinu =-3_/5 , u in QIII, sinv=-12_/13,vin QIV

in QIII, cosu is negative , in QIV cosv is positive

sin²x+cos²x =1 is an identity

so

sin²u+cos²u =1

(-3_/5)²+cos²u =1

(9_/25)+cos²u =1

cos²u =1-(9_/25)

cos²u =(16_/25)

cosu =(-4_/5)

sin²v+cos²v =1

(-12_/13)²+cos²v =1

(144_/169)+cos²v =1

cos²v =1-(144_/169)

cos²v =(25_/169)

cosv =(5_/13)

cos(u+v) =cosucosv -sinusinv

cos(u+v) =((-4_/5)(5_/13))- ((-3_/5)(-12_/13))

cos(u+v) =(-20_/65)- (36_/65)

cos(u+v) =-56_/65