Find all solutions of cos (x) + 1/2 sec (x) = -3/2 in the interval (2pi, 4pi)

Question

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Answer 1

cosx + (1/2)(1/cosx) = -3/2
times 2
2cosx + 1/cosx = -3
times cosx
2cos^2 x + 1 = -3cosx
2cos^2 x + 3cosx + 1 = 0

(2cosx + 1)(cosx + 1) = 0
cosx = -1/2 or cosx = -1

x = 120° or x = 240° or x = 180°
in radians:
x = 2π/3 , 4π/3 , π

BUT, you want the solutions in (2π,4π) so let's just look one period over
x = 2π/3 + 2π = 8π/3
x = 4π/3 + 2π = 10π/3
x = π + 2π = 3π