Find (a) sin(α + β), (b) cos(α + β), and (c) the quadrant containing α + β.

Question

tan α = 3/10, π < α <3/2π; tan β = −5/4, 1/2π < β < π

Answer 1

tan α = 3/10, π < α <3/2π; tan β = −5/4, 1/2π < β < π

  just adding a and b ranges find 3π/2<a+b<5π/2

a+b either in Q4 or Q1

cos(a)=-10/√109,,,,,sin(a)=-3/√109

cos(b)=-4/√41,,,,,sin(b)=5/√41

doing sin first and whether it is positive or negative will resolve quadrant

sin(a+b)=([-3*-4]/√[109*41])+([5*-10]?√(109*41])<0 so a+b is in Q4

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