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

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Answer 1

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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.?

Cos(beta) = 3/5

Both reference triangles are 3,4,5. 

Alpha is in Q3 where sin & cos are both negative. 

sin(alpha) = -4/5

cos(alpha) = -3/5

tan(alpha) = 4/3

Beta is in Q1 where sin & cos are both positive. 

sin(beta) = 4/5

cos(beta) = 3/5

tan(beta) = 4/3

sin(alpha+beta) 

= (-4/5)(3/5) + (-3/5)(4/5)

= -24/25

tan(alpha+beta)

= [(4/3) + (4/3)]/[1-(4/3)(4/3)]

= (8/3)/(-7/9)

= -(9*8)/(7*3) = -(3*8)/7 = - 24/7

for (alpha+beta), sin and tan are both negative. This puts the angle in Q4. 

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Answer 2

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