# URGENT PRE-CALCULUS HELP!?

sin α = 4/5, π/2 < α < π
cos β = 5/13, 0 < β < π
Find cos (α + β) , tan (α - β) , sin2α , and cos(β/2)

Please show work! This is the last part of an assignment I have and it's the only part I don't get!

sin α = 4/5, π/2 < α < π
cos β = 5/13, 0 < β < π

sin² α + cos² α =1

(4/5)² +  cos² α =1

cos² α = 1 - (16/25)

cos² α =  9/25

cos  α =  3/5

sin² β + cos² β =1

sin² β + (5/13)² = 1

sin² β =  1 - 25/169

sin² β =  144/169

sin β =   12/13

1)  cos(α + β) = cosα cosβ - sinα sinβ

= (3/5)(5/13) - (4/5)(12/13)

=  15/65 - 48/65

=  -33/65

cos(α + β)  =   -33/65

2)  tan (α - β)

tan α = sinα/ cosα

= (4/5)/(3/5)

= 4/3

tan β  = sinβ/ cosβ

=  (12 /13)/(5/13)

=  12/5

tan (α - β) = (tanα - tanβ)/(1- tanα tanβ)

= (4/3 - 12/5)/(1- (4/3)(12/5) )

=  (-16/15) / (1- 16/5)

=  (-16/15) / (-11/5)

=  16/33

tan (α - β) =  16/33

3)  sin2α

sin2α =  2 sinα cosα

=  2 (4/5) (3/5)

=  24/25

sin2α  = 24/25

4) cos(β/2)

cos (β/2) = ±√(1+ cos β)/2

= ± √(1+ 5/13)/2

= ± √(18/13)/2

= ± √18/26

cos (β/2)   = ± √18/26