Answer each question.


Suppose solving a trigonometric equation for solutions over the interval [0, 2π) leads to 2x = 2π/3, 2π, 8π/3. What are the corresponding values of x?

Verified step by step guidance

1. Start with the given solutions for 2x: \(2x = \frac{2\pi}{3}, 2\pi, \frac{8\pi}{3}\). Our goal is to find the corresponding values of \(x\) over the interval \([0, 2\pi)\).
2. To isolate \(x\), divide each equation by 2: \(x = \frac{2\pi}{3} \div 2, \quad x = 2\pi \div 2, \quad x = \frac{8\pi}{3} \div 2\).
3. Perform the division for each term: \(x = \frac{2\pi}{3} \times \frac{1}{2} = \frac{2\pi}{6} = \frac{\pi}{3}\), \(x = \frac{2\pi}{2} = \pi\), and \(x = \frac{8\pi}{3} \times \frac{1}{2} = \frac{8\pi}{6} = \frac{4\pi}{3}\).
4. Check that each value of \(x\) lies within the interval \([0, 2\pi)\). Since \(\frac{\pi}{3}, \pi, \frac{4\pi}{3}\) are all between 0 and \(2\pi\), they are valid solutions.
5. Thus, the corresponding values of \(x\) are \(\frac{\pi}{3}, \pi, \frac{4\pi}{3}\).

Key Concepts

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angle values that satisfy the given equation within a specified interval. This often requires isolating the trigonometric function and considering the periodic nature of trigonometric functions to find all possible solutions.

Interval Restriction and Solution Sets

When solving equations over a specific interval, such as [0, 2π), solutions must be adjusted or filtered to lie within that range. This means any solution outside the interval should be modified by subtracting or adding multiples of the period (2π) to bring it within the interval.

Inverse Operations and Division in Equations

If the equation involves a multiple of the variable (e.g., 2x), solving for x requires dividing both sides by that multiple. Care must be taken to apply this operation correctly to each solution to find the corresponding values of x.