Consider the equation $az^2 + z + 1 = 0$ having purely imaginary root where $a = \cos \theta + i \sin \theta, i = \sqrt{-1}$ and function $f(x) = x^3 - 3x^2 + 3(1 + \cos \theta)x + 5,$ then answer the following questions. Number of roots of the equation $\cos 2\theta = \cos \theta,\theta \in [0, 4\pi]$ are

Using the double angle formula, write cos 2 theta as 2 cos^2 theta - 1. The equation becomes 2 cos^2 theta - cos theta - 1 = 0, which factors into (2 cos theta + 1)(cos theta - 1) = 0. This gives cos theta = -1/2 or cos theta = 1. In the interval [0, 4 pi], the equation cos theta = -1/2 yields the three solutions 2 pi/3, 4 pi/3, and 8 pi/3, while cos theta = 1 yields the additional solution 2 pi. Counting these gives a total of 4 roots.

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