Solve each equation for exact solutions over the interval [0, 2π).
cos² x + 2 cos x + 1 = 0

Guida verificata passo dopo passo

Recognize that the equation is a quadratic in terms of \( \cos x \). Rewrite the equation as \( (\cos x)^2 + 2 \cos x + 1 = 0 \).

Notice that the quadratic expression can be factored as \( (\cos x + 1)^2 = 0 \).

Set the factor equal to zero: \( \cos x + 1 = 0 \), which simplifies to \( \cos x = -1 \).

Recall the unit circle values where \( \cos x = -1 \) within the interval \( [0, 2\pi) \).

Identify the exact solution(s) for \( x \) where \( \cos x = -1 \) in the given interval.

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Quadratic Form in Trigonometric Equations

Many trigonometric equations can be rewritten as quadratic equations by substituting a trigonometric function (e.g., cos x) with a variable. This allows the use of algebraic methods like factoring or the quadratic formula to find solutions.

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Solving for Cosine Values

After rewriting the equation in quadratic form, solving for cos x involves finding the roots of the quadratic. These roots represent the cosine values for which the original equation holds true, which can then be used to find the corresponding angles.

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Finding Exact Solutions on the Interval [0, 2π)

Once the cosine values are found, the exact solutions for x are determined by identifying all angles within the interval [0, 2π) whose cosine matches those values. This often involves using the unit circle and understanding cosine symmetry.

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