Solve each equation for x, where x is restricted to the given interval.
y = 6 cos x/4 , for x in [0, 4π]

Verified step by step guidance

1. Step 1: Start by isolating the cosine function. Divide both sides of the equation by 6 to get \( \cos\left(\frac{x}{4}\right) = \frac{y}{6} \).
2. Step 2: Use the inverse cosine function to solve for \( \frac{x}{4} \). This gives \( \frac{x}{4} = \cos^{-1}\left(\frac{y}{6}\right) \).
3. Step 3: Multiply both sides of the equation by 4 to solve for \( x \). This results in \( x = 4 \cdot \cos^{-1}\left(\frac{y}{6}\right) \).
4. Step 4: Consider the periodic nature of the cosine function. Since \( \cos(\theta) = \cos(2\pi - \theta) \), the general solution for \( x \) is \( x = 4 \cdot \cos^{-1}\left(\frac{y}{6}\right) + 8k\pi \) and \( x = 4 \cdot (2\pi - \cos^{-1}\left(\frac{y}{6}\right)) + 8k\pi \), where \( k \) is an integer.
5. Step 5: Determine the values of \( k \) such that \( x \) is within the interval \([0, 4\pi]\). Calculate the specific values of \( x \) that satisfy the equation within this interval.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle. It is periodic with a period of 2π, meaning it repeats its values every 2π units. Understanding the behavior of the cosine function is essential for solving equations involving it, particularly in determining the values of x that satisfy the equation within a specified interval.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy a given equation. This often requires using inverse trigonometric functions and understanding the periodic nature of trigonometric functions. In this case, we need to isolate x and consider the periodic solutions within the interval [0, 4π], which may yield multiple valid solutions due to the cosine function's periodicity.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [0, 4π] indicates that x can take any value from 0 to 4π, inclusive. Understanding how to interpret and work within this interval is crucial for determining the valid solutions to the equation, as it restricts the possible values of x to a specific range.