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explain the steps needed to find the points for x when the y value is 1/2 after sketching this graph

To address your question regarding how to find the points \( x \) when the \( y \)-value is \( \frac{1}{2} \) after sketching the graph of \( f(x) = \cos x \) over the interval \( [-2\pi, 2\pi] \), here are the steps:

1. **Understand the Function and its Domain**:
   - The function given is \( f(x) = \cos x \).
   - The domain is restricted to \( [-2\pi, 2\pi] \).
   - The cosine function is periodic with a period of \( 2\pi \), so we expect a repeating pattern within this interval.

2. **Identify the Points Where \( \cos x = \frac{1}{2} \)**:
   - We solve the equation \( \cos x = \frac{1}{2} \).
   - The general solution for \( \cos x = \frac{1}{2} \) in one period is \( x = \pm \frac{\pi}{3} + 2k\pi \), where \( k \) is any integer representing the periodic nature of the cosine function.

3. **Find the Specific Solutions within the Interval**:
   - Since the interval is \( [-2\pi, 2\pi] \), we need to find the values of \( x \) that lie in this interval.
   - Starting with the general solution:
     - For \( k = 0 \): \( x = \frac{\pi}{3}, -\frac{\pi}{3} \)
     - For \( k = -1 \): \( x = \frac{\pi}{3} - 2\pi = -\frac{5\pi}{3} \)
     - For \( k = 1 \): \( x = \frac{\pi}{3} + 2\pi = \frac{7\pi}{3} \), but this value is outside the given interval, so it doesn't count.

   Therefore, the solutions are \( x = -\frac{5\pi}{3}, -\frac{\pi}{3}, \frac{\pi}{3}, \frac{5\pi}{3} \).

4. **Sketch the Graph**:
   - Draw the cosine wave between \( -2\pi \) and \( 2\pi \). The graph should have a peak at \( y = 1 \) at \( x = 0 \) and symmetric troughs at \( y = -1 \) at \( x = \pm \pi \).
   - Mark the points where the graph crosses \( y = \frac{1}{2} \) at the calculated \( x \)-coordinates: \( -\frac{5\pi}{3}, -\frac{\pi}{3}, \frac{\pi}{3}, \frac{5\pi}{3} \).

By following these steps, you can find the points where \( \cos x = \frac{1}{2} \) after sketching the graph.

Would you like further clarification on any step?

Here are 5 related questions:
1. What is the period of the cosine function, and how does it affect the graph?
2. How would the solutions change if the function were \( \cos(2x) \) instead of \( \cos x \)?
3. What happens to the graph if we shift it by a phase, such as \( f(x) = \cos(x - \frac{\pi}{4}) \)?
4. How does the amplitude affect the points where \( y = \frac{1}{2} \)?
5. How would you find the points where \( \cos x = -\frac{1}{2} \)?

**Tip**: For periodic functions like sine and cosine, understanding the general solution and the behavior of the function across its period is key to finding specific points.

Learn how to find the points where y = 1/2 on the graph of f(x) = cos(x) within the interval [-2π, 2π]. This guide covers the steps to solve trigonometric equations, understand periodic functions, and sketch cosine graphs. Suitable for students in Grades 10-12.

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