Question

what is the x-intercept of y=sin x

Answer 1

how can the x-intercept be written out?

Answer 2

To find the x-intercepts of the function \( y = \sin x \), you need to determine where the function equals zero. This occurs when:

\[
\sin x = 0
\]

The sine function is equal to zero at integer multiples of \( \pi \). Therefore, the x-intercepts are given by:

\[
x = n\pi
\]

where \( n \) is any integer (i.e., \( n = 0, \pm 1, \pm 2, \ldots \)).

So, the x-intercepts of \( y = \sin x \) are:

\[
x = 0, \pm \pi, \pm 2\pi, \pm 3\pi, \ldots
\]

Answer 3

The x-intercepts of the function \( y = \sin x \) can be written explicitly as a set of points. Since the x-intercepts occur at \( x = n\pi \) for any integer \( n \), you can write them out as:

\[
\{ (n\pi, 0) \mid n \in \mathbb{Z} \}
\]

Where \( \mathbb{Z} \) represents the set of all integers (positive, negative, and zero).

If you want to list the first few x-intercepts, they are:

- At \( n = 0 \): \( (0, 0) \)
- At \( n = 1 \): \( (\pi, 0) \)
- At \( n = -1 \): \( (-\pi, 0) \)
- At \( n = 2 \): \( (2\pi, 0) \)
- At \( n = -2 \): \( (-2\pi, 0) \)

And so forth. Thus, the x-intercepts can be expressed as:

\[
\{ (0, 0), (\pi, 0), (-\pi, 0), (2\pi, 0), (-2\pi, 0), (3\pi, 0), (-3\pi, 0), \ldots \}
\]

This illustrates the repeated nature of the x-intercepts along the x-axis at every integer multiple of \( \pi \).