Question

If cos2θ + 3cosθ + 2 = 0, then cosθ = _____

Answer 1

Hello,

cos θ =-1 or cos θ =-1/2

Indeed:
cos 2θ =2 cos² θ -1

cos 2θ +3 cos θ+2=0
==>2 cos² θ -1+3 cos θ+2=0
==>2 cos² θ +3 cos θ+1=0
Δ=3²-4*2=1
cos θ =(-3-1)/4=-1 or cos θ =(-3+1)/4=-1/2


We must verify:

if cos θ =-1 then 2*(-1)² -1 +3*(-1)+2=2-1-3+2=0

if cos θ =-1/2 then 2*(-1/2)² -1 +3*(-1/2)+2=2*1/4-1-3/2+2=1/2-3/2+1=0

Answer 2

cos²θ + 3cosθ + 2 = 0
cos² + cosθ + 2cosθ + 2 = 0
cosθ(cosθ) + cosθ(1) + 2(cosθ) + 2(1) = 0
cosθ(cosθ + 1) + 2(cosθ + 1) = 0
(cosθ + 2)(cosθ + 1) = 0
cosθ + 2 = 0 or cosθ + 1 = 0
- 2 - 2 - 1 - 1
cosθ = -2 cosθ = -1
cos⁻¹(cosθ) = cos¹(-1)
θ = 180