Diketahui: sin alpha=3/5 dan sin beta=8/17, alpha adalah

a. tan(α+β) = 13/84, b. tan(α-β) = 77/36

Pembahasan

Diketahui:
sin α = 3/5 (α lancip)
sin β = 8/17 (β tumpul)

a. Hitung tan(α+β)
Pertama, kita perlu mencari nilai cos α, tan α, cos β, dan tan β.
Karena α lancip:
cos α = √(1 - sin^2 α) = √(1 - (3/5)^2) = √(1 - 9/25) = √(16/25) = 4/5
tan α = sin α / cos α = (3/5) / (4/5) = 3/4

Karena β tumpul:
cos β = -√(1 - sin^2 β) = -√(1 - (8/17)^2) = -√(1 - 64/289) = -√(225/289) = -15/17
tan β = sin β / cos β = (8/17) / (-15/17) = -8/15

Rumus tan(α+β) = (tan α + tan β) / (1 - tan α tan β)
tan(α+β) = (3/4 + (-8/15)) / (1 - (3/4)(-8/15))
tan(α+β) = ((45 - 32) / 60) / (1 + 24/60)
tan(α+β) = (13/60) / ((60 + 24) / 60)
tan(α+β) = (13/60) / (84/60)
tan(α+β) = 13/84

b. Hitung tan(α-β)
Rumus tan(α-β) = (tan α - tan β) / (1 + tan α tan β)
tan(α-β) = (3/4 - (-8/15)) / (1 + (3/4)(-8/15))
tan(α-β) = ((45 + 32) / 60) / (1 - 24/60)
tan(α-β) = (77/60) / ((60 - 24) / 60)
tan(α-β) = (77/60) / (36/60)
tan(α-β) = 77/36

Jadi:
a. tan(α+β) = 13/84
b. tan(α-β) = 77/36