Question

Question
At which values of x does the graph of f left parenthesis x right parenthesis equals cosine x intersect the x-axis? Select all that apply.
(1 point)
Responses

0
0

Start Fraction pi over 2 End Fraction
Image with alt text: Start Fraction pi over 2 End Fraction

π
π

Start Fraction 3 pi over 2 End Fraction
Image with alt text: Start Fraction 3 pi over 2 End Fraction

2π

Answer 1

The graph of the function \( f(x) = \cos(x) \) intersects the x-axis at the points where the function equals zero, i.e., when \( \cos(x) = 0 \).

The cosine function is zero at the angles:

\[
x = \frac{\pi}{2} + k\pi
\]

where \( k \) is any integer. Specifically, within the interval \( [0, 2\pi] \), the values at which \( \cos(x) = 0 \) are:

1. \( x = \frac{\pi}{2} \)
2. \( x = \frac{3\pi}{2} \)

From the options provided:

- \( 0 \) — \( \cos(0) = 1 \), does not intersect the x-axis.
- \( \frac{\pi}{2} \) — \( \cos\left(\frac{\pi}{2}\right) = 0\), intersects the x-axis.
- \( \pi \) — \( \cos(\pi) = -1 \), does not intersect the x-axis.
- \( \frac{3\pi}{2} \) — \( \cos\left(\frac{3\pi}{2}\right) = 0\), intersects the x-axis.
- \( 2\pi \) — \( \cos(2\pi) = 1 \), does not intersect the x-axis.

Thus, the values of \( x \) where the graph of \( f(x) = \cos(x) \) intersects the x-axis are:

- \( \frac{\pi}{2} \)
- \( \frac{3\pi}{2} \)

So, the correct responses are:
- \( \frac{\pi}{2} \)
- \( \frac{3\pi}{2} \)