Let 
f(x) = cos(x).
 Find the x-intercepts of f(x) on 
[0, 2𝜋).       find the smaller x value and larger x value

The x-intercepts of the function \( f(x) = \cos(x) \) are the points where the function equals zero, i.e., where:

\[
\cos(x) = 0
\]

We need to find these values of \( x \) within the interval \([0, 2\pi)\).

### Step-by-Step Process:
1. The cosine function equals zero at:
   \[
   x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}
   \]
2. We need to find the specific values of \( x \) that lie within the interval \([0, 2\pi)\).

Let's calculate these:

- For \( n = 0 \):
  \[
  x = \frac{\pi}{2}
  \]

- For \( n = 1 \):
  \[
  x = \frac{3\pi}{2}
  \]

So, the x-intercepts of \( f(x) = \cos(x) \) on the interval \([0, 2\pi)\) are:

- The smaller \( x \)-value: \( \frac{\pi}{2} \)
- The larger \( x \)-value: \( \frac{3\pi}{2} \)

### Conclusion:
- The smaller x-intercept: \( \frac{\pi}{2} \)
- The larger x-intercept: \( \frac{3\pi}{2} \)

Would you like further details, or have any other questions?

Here are 5 related questions:
1. What are the general x-intercepts of the function \( \cos(x) \)?
2. How does the period of the cosine function affect the intercepts?
3. Can you find the intercepts for \( \cos(x) = 0 \) on the interval \( [0, 4\pi) \)?
4. What is the amplitude and period of the function \( f(x) = \cos(x) \)?
5. What happens to the x-intercepts if the function is \( f(x) = \cos(2x) \)?

**Tip:** When working with trigonometric functions, always check the periodicity to find all possible solutions within a given interval.

Let f(x) = cos(x). Find the x-intercepts of f(x) on [0, 2𝜋). Find the smaller x value and larger x value.

Discover how to find the x-intercepts of the cosine function f(x) = cos(x) on the interval [0, 2π). This problem explains the key trigonometric concepts and formulas, including periodicity and solving for x when cos(x) = 0. Suitable for high school students in Grades 9-12.

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