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.