Find exact solutions

Question

find exact solutions on the interval 0≤θ≥2Π. 4cos²θ + 4cosθ - 3 =0

Answer 1

The expression on the left hand side can be factored so:

 

   (2 cos(θ) -1) ( 2 cos(θ) +3)  = 0

 

  Thus either 2 cos(θ) -1 =0     or   2 cos(θ) +3 =0

  but there are no solutions for   2 cos(θ) + 3 = 0    so

 

      2 cos(θ) - 1 = 0     so   cos(θ) =   1/2

    The exact solutions to this (in radian measure) can be found on a standard unit circle chart.  They are

 

    θ = pi/3    and  θ =  5 pi/3