Solutions of the equation

Question

Find all solutions of the equation 2cos^2(t)-cos(t)-1=0 that lie in the interval [0,2pi) I found two solutions 2pi/3 and 4pi/3, but I'm having trouble finding a third. 

Answer 1

Let x = cos t, so your equation becomes 2x² - x - 1 = 0, or (2x + 1)(x - 1) = 0. The roots are x = -1/2, 1. So cos t = -1/2, 1. The solutions you found are the solutions to cos t = -1/2 in [0, 2π). The third solution is t = 0, which comes from solving cos t = 1 in [0, 2π).