If sin α = − 3/5 and sec β = 5/4 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

Alex:

Use sin(α+β)=sin(α)*cos(β)+sin(β)*cos(α) (expansion 1)

tan(α+β) = (tan(α)+tan(β))/(1-tan(α)*tan(β)) (expansion 2)

We know sin(α) = -3/5 and α is in the third quadrant. In the third quadrant, sin < 0, cos < 0, and tan >0

cos(α) = -4/5 and tan(α) = 3/4 try to understand why.

We also know sec(β)=5/4 and β is in the first quadrant. In the first quadrant, sin >0, cos >0, and tan > 0

cos(β) = 1/sec(β) = 4/5 sin(β) = 3/5 and tan(β) = 3/4.

Plug these values into the expansions 1 and 2. Let me know if you need any help!