solve the equation in the interval [0, 2pi)
cos^2 x - cos x - 1 = 0
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To solve the equation cos^2 x - cos x - 1 = 0 in the interval [0, 2π), we can use the quadratic formula or factorization method. Let's use factorization in this case.
Step 1: Rewrite the equation as (cos x - 1)(cos x + 1) = 0.
Step 2: Set each factor equal to zero and solve for x:
cos x - 1 = 0 --> cos x = 1
cos x + 1 = 0 --> cos x = -1
Step 3: Considering the interval [0, 2π), we need to find the values of x where cos x equals 1 or -1 within this interval.
For cos x = 1, we know that x = 0 since cos 0 = 1.
For cos x = -1, we know that x = π since cos π = -1.
Therefore, the solutions to the equation sin^2 x - sin x - 1 = 0 in the interval [0, 2π) are x = 0 and x = π.