find sin alpha+beta given sin alpha and tan beta

Question

Answer 1

sin α =15/17  -3π/2<α<-π

α is in 2nd quadrant.        

In 2nd quadrant sin cosecant are positive.

sin α = opp/hyp =15/17

opp =15 and hyp =17

adj = sqrt(17^2-15^2) =sqrt(289-225)=8

cos α= -8/17

tanβ=-√5          π/2<α<-π

β is in 2nd quadrant.        

In 2nd quadrant sin cosecant are positive.

tan β = opp/adj =-√5

opp =√5 and adj =1

hyp = sqrt((√5)^2+1^2) =sqrt(6)

sin β= √5/√6.cos β=- 1/√6

a)

sin(α+β) =sinαcosβ+cosαsinβ

              =(15/17)(- 1/√6)+(-8/17)(√5/√6)

              =(-15-8√5)/(17√6)

b)

cos(α+β) =cosαcosβ-sinαsinβ

              =(-8/17)(- 1/√6)-(15/17)(√5/√6)

              =(8-15√5)/(17√6)

c) sin(α-β) =sinαcosβ-cosαsinβ

              =(15/17)(- 1/√6)-(-8/17)(√5/√6)

              =(-15+8√5)/(17√6)

d) cos(α-β) =cosαcosβ+sinαsinβ

              =(-8/17)(- 1/√6)+(15/17)(√5/√6)

              =(8+15√5)/(17√6)

tan(α-β) =sin(α-β) /cos(α-β)

              =[(-15+8√5)/(17√6)]/(8+15√5)/(17√6)

              =[(-15+8√5)/[8+15√5]