Solve each equation for all exact solutions, in radians.
cos 2x + cos x = 0

Verified step by step guidance

1. Start with the given equation: \(\cos 2x + \cos x = 0\).
2. Use the double-angle identity for cosine: \(\cos 2x = 2\cos^2 x - 1\). Substitute this into the equation to get \(2\cos^2 x - 1 + \cos x = 0\).
3. Rewrite the equation as a quadratic in terms of \(\cos x\): \(2\cos^2 x + \cos x - 1 = 0\).
4. Solve the quadratic equation \$2y^2 + y - 1 = 0\( where \(y = \cos x\), using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \)a=2\(, \)b=1\(, and \)c=-1$.
5. For each solution \(y = \cos x\), find all values of \(x\) in radians that satisfy \(\cos x = y\), considering the periodicity of cosine and including all exact solutions.

Key Concepts

Double-Angle Identity for Cosine

The double-angle identity expresses cos(2x) in terms of cos(x) or sin(x). Common forms include cos(2x) = 2cos²(x) - 1 or cos(2x) = 1 - 2sin²(x). This identity helps rewrite the equation to a single trigonometric function, simplifying the solving process.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within the domain. Since trigonometric functions are periodic, solutions repeat every 2π or π, so general solutions include these periodic terms.

Factoring Trigonometric Expressions

Factoring is a method to simplify trigonometric equations by expressing them as a product of factors set to zero. This allows using the zero-product property to find multiple solutions by solving each factor separately.