For the function [tex]\( y = \cos(x) \)[/tex], we are interested in finding the [tex]\( x \)[/tex]-coordinates where the function reaches its maximum values.
The cosine function, [tex]\( \cos(x) \)[/tex], reaches its maximum value of 1 at certain points along the [tex]\( x \)[/tex]-axis. The maximum values occur at angles where the cosine curve reaches its peaks, which is exactly at integer multiples of [tex]\( 2\pi \)[/tex]:
[tex]\[ x = 2k\pi \][/tex]
where [tex]\( k \)[/tex] is any integer. This means that at [tex]\( k = 0, \pm 1, \pm 2, \pm 3, \ldots \)[/tex], the function [tex]\( \cos(x) \)[/tex] will achieve its maximum value.
If we restrict the integer [tex]\( k \)[/tex] to be 0, and even integers (which are multiples of 2), we can simplify the formula to:
[tex]\[ x = k\pi \][/tex]
where [tex]\( k \)[/tex] takes values such as [tex]\( 0, \pm 2, \pm 4, \ldots \)[/tex].
Since these points are straightforward and easier to write in the form of [tex]\( \pi \)[/tex], our practical and accurate expression for the [tex]\( x \)[/tex]-coordinates of the maximum values would be:
[tex]\[ x = k\pi \][/tex]
where [tex]\( k = 0, \pm 2, \pm 4, \ldots \)[/tex]
So, the correct choice is:
[tex]\[ k \pi \text{ for } k=0, \pm 2, \pm 4, \ldots. \][/tex]
