If tan α = − 21/20 and cot β = 12/35 for a second-quadrant angle α and a third-quadrant angle β, find the following.

Question

 sin(α + β)

cos(α + β)

 tan(α + β)

Answer 1

For tan(a)=-21/20 in the 2nd quadrant, we can determine the hypotenuse of the respective right triangle to be sqrt(21^2 + 20^2) = 29. With this, we know that sin(a) = 21/29 and cos(a) = -20/29 because sine is positive in the 2nd quadrant and cosine is negative. For cot(b) = 12/25 in the 3rd quadrant, the hypotenuse is sqrt(12^2 + 35^2) = 37. Thus, sin(b) = -35/37 and cos(b) = -12/37 because sine and cosine are both negative in the 3rd quadrant.

sin(a) = 21/29 ; cos(a) = -20/29

sin(b) = -35/37 ; cos(b) = -12/37

Formulas:

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

tan(a+b) = sin(a+b) / cos(a+b)

Plugging in:

sin(a+b) = (21/29)*(-12/37) + (-20/29)*(-35/37) = 448/1073

cos(a+b) = (-20/29)(-12/37) - (21/29)(-35/37) = 975/1073

tan(a+b) = (448/1073) / (975/1073) = 448/975