In the triangle △ABC, introduce the notations a=|BC|, b=|CA|, c=|AB|, as well as ∠A=α(alpha), ∠B=β (beta) and ∠C=γ (gamma). Find b, given that a=5, sinα(alpha)=2/3 and cos β=5/8. b=?

Answer 1

Use the Law of Sines:

sin(α)/5 = sin(β)/b

bsin(α) = 5sin(β)

b = 5sin(β)/sin(α)

Since you are lacking sin(β), use the information given that cos(β) = 5/8:

Since cos(β) = adjacent/hypotenuse, this sketch holds true:

To find side "x", use the Pythagorean Theorem. 5² + x² = 8² so x = √(64-25) = √39 therefore sin(β) = √39/8

b = 5sin(β)/sin(α)

b = (5)(√39/8)/(2/3) = 15√39/16 which is approx 5.855