If 𝛼 and 𝛽 are acute angles such that csc 𝛼 = 5 /3 and cot 𝛽 = 8 /15 , find the following.

Question

If 𝛼 and 𝛽 are acute angles such that csc 𝛼 = 5 /3 and cot 𝛽 = 8 /15 , find the following.

(a)    sin(𝛼 + 𝛽)

(b)    tan(𝛼 + 𝛽)

(c) The quadrant containing 𝛼 + 𝛽 is

Answer 1

(c) quadrant I for alpha or beta. alpha + beta could be I or II. but it's quadrant II because tan(alpha+beta)<0

(a) sinalpha = 3/5, cosalpha =4/5

sinbeta = =15/sqr(8^2+15^2) =15/17, cosbeta = 8/17

sin(alpha+beta) = sin(alpha)cos(beta) + sin(beta)cos(alpha) = (3/5)(8/17)+(15/17)(4/5) = 84/85

(c) tan(alpha+beta) = sin(alpha+beta)/cos(alpha+beta)

cos(alpha+beta) = cosalpha(cosbeta) = sinalpha(sinbeta) = -13/85

tan(alpha+beta) = 84/85 divided by - 13/85 = - 84/13

Answer 2

sin∝ = 1/csc∝= 3/5; cos∝ = √1 - (3/5)² = 4/5; tan∝ = sin∝/cos∝ = (3/5)/(4/5) = 3/4

tanβ = 1/cotβ = 15/8; secβ = √1 + (15/8)² = 17/8; cosβ = 8/17, sinβ = √1 - (8/17)² = 15/17

(a) sin(α + β)= sinαcosβ + cosαsinβ = (3/5)(8/17) + (4/5)(15/17) = 24/85 + 60/85 = 84/85

(b) tan(α + β) = (tanα + tanβ)/(1 - tanαtanβ) = (3/4 + 15/8)/(1 - 3/4·15/8) = 21/8/(- 13/32) = - 84/13

(c) Because α and β are acute angles and tan(α + β) = -84/13 < 0, then (α + β) in quadrant II.