Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). cos 4x = ﹣√3 / 2

Verified step by step guidance

1. Identify the given equation: \(\cos 4x = -\frac{\sqrt{3}}{2}\).
2. Recall the general solutions for \(\cos \theta = -\frac{\sqrt{3}}{2}\), which occur at angles where the cosine value is \(-\frac{\sqrt{3}}{2}\). These angles in \([0, 2\pi)\) are \(\theta = \frac{5\pi}{6}\) and \(\theta = \frac{7\pi}{6}\).
3. Set \$4x\( equal to each of these angles plus the general solution for cosine, which repeats every \(2\pi\): \(4x = \frac{5\pi}{6} + 2k\pi\) and \(4x = \frac{7\pi}{6} + 2k\pi\), where \)k$ is any integer.
4. Solve each equation for \(x\) by dividing both sides by 4: \(x = \frac{5\pi}{24} + \frac{k\pi}{2}\) and \(x = \frac{7\pi}{24} + \frac{k\pi}{2}\).
5. Find all values of \(x\) within the interval \([0, 2\pi)\) by substituting integer values of \(k\) such that \(x\) remains in this interval.

Key Concepts

Multiple-Angle Trigonometric Equations

These are equations where the trigonometric function's argument is a multiple of the variable, such as cos(4x). Solving them requires understanding how to handle the increased frequency and finding all solutions within the given interval by adjusting for the multiple angle.

Inverse Trigonometric Functions and Principal Values

To solve equations like cos(4x) = -√3/2, we use inverse cosine to find principal angle solutions. Since cosine is periodic and even, multiple solutions exist within [0, 2π), so all possible angles that satisfy the equation must be considered.

Interval Restriction and Solution Set

The problem restricts solutions to the interval [0, 2π). Because the argument is 4x, the period is shortened, leading to multiple solutions within this interval. After finding general solutions, we must adjust and filter them to fit the original interval for x.