In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). cos 2x = cos x

Verified step by step guidance

1. Recall the double-angle identity for cosine: \(\cos 2x = 2\cos^{2} x - 1\). Substitute this into the equation to rewrite it as \(2\cos^{2} x - 1 = \cos x\).
2. Rearrange the equation to set it equal to zero: \(2\cos^{2} x - \cos x - 1 = 0\).
3. Let \(y = \cos x\) to transform the equation into a quadratic form: \$2y^{2} - y - 1 = 0$.
4. Solve the quadratic equation \$2y^{2} - y - 1 = 0\( for \)y\( using the quadratic formula \(y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \)a=2\(, \)b=-1\(, and \)c=-1$.
5. For each solution \(y\), find the corresponding values of \(x\) in the interval \([0, 2\pi)\) by solving \(\cos x = y\). Use the unit circle or inverse cosine function to determine all valid solutions.

Key Concepts

Double-Angle Identity for Cosine

The double-angle identity expresses cos(2x) in terms of cos(x) and sin(x). Common forms include cos(2x) = 2cos²(x) - 1 or cos(2x) = 1 - 2sin²(x). This identity allows rewriting the equation cos(2x) = cos(x) into a solvable form involving a single trigonometric function.

Solving Trigonometric Equations

Solving trigonometric equations involves manipulating the equation using identities and algebraic techniques to isolate the variable. After rewriting, solutions are found by considering the unit circle values where the trigonometric functions equal specific values within the given interval.

Interval Restriction and General Solutions

When solving trigonometric equations, solutions must be restricted to the specified interval, here [0, 2π). This means identifying all angles within one full rotation of the unit circle that satisfy the equation, ensuring the final answer fits the problem’s domain.