If sin 𝛼 = − 4/5 and sec 𝛽 = 5/3 for a third-quadrant angle 𝛼 and a first-quadrant angle 𝛽, find the following.

Question

(A) sin(𝛼 + 𝛽)

(B) tan(𝛼 + 𝛽)

Answer 1

sin(𝛼 + 𝛽) = sinαcosβ+sinβcosα tan(𝛼 + 𝛽) = (tanα + tanβ) / (1 - tanαtanβ)

QIII: sinα = - 4/5 , cosα = - 3/5 , tanα = 4/3

QI : secβ = 5/3 , cosβ = 3/5 , sinβ = 4/5 , tanβ = 4/3

sin(𝛼 + 𝛽) = (-4/5)(3/5) + (4/5)(-3/5) = -24/25

tan(𝛼 + 𝛽) = (8/3) / (1 - 16/9) = (8/3)(-7/9) = -24/7

Notice how the sum and difference formulas for trig functions generate pythagorean triplet triangles given two pythagorean triplet triangles initially.