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. Identify the given equation: \(y = 6 \cos \frac{x}{4}\), and the interval for \(x\) is \([0, 4\pi]\).
2. Since the equation is in terms of \(y\), decide what value of \(y\) you want to solve for. For example, if you want to find \(x\) when \(y\) equals a specific value, set \(6 \cos \frac{x}{4} = y_0\) where \(y_0\) is that value.
3. Isolate the cosine term by dividing both sides by 6: \(\cos \frac{x}{4} = \frac{y_0}{6}\).
4. Use the inverse cosine function to solve for \(\frac{x}{4}\): \(\frac{x}{4} = \arccos \left( \frac{y_0}{6} \right)\).
5. Multiply both sides by 4 to solve for \(x\): \(x = 4 \arccos \left( \frac{y_0}{6} \right)\). Remember to consider all solutions for \(x\) within the interval \([0, 4\pi]\) by using the periodicity and symmetry of the cosine function.

Key Concepts

Solving Trigonometric Equations

Solving trigonometric equations involves finding all values of the variable that satisfy the equation within a specified interval. This often requires isolating the trigonometric function and using inverse functions or known values of sine, cosine, or tangent to determine solutions.

Properties of the Cosine Function

The cosine function is periodic with a period of 2π, meaning its values repeat every 2π units. Understanding its graph, symmetry, and key values helps in identifying all solutions within a given interval, especially when the argument of cosine is scaled or shifted.

Interval Restrictions and Domain Considerations

When solving equations on a restricted interval, it is essential to consider only solutions that lie within that domain. This involves adjusting for the function's period and ensuring that all valid solutions for x fall within the specified range, such as [0, 4π] in this problem.