To determine the value of [tex]\( x \)[/tex] such that [tex]\(\sin(x) = \cos(32^\circ)\)[/tex], where [tex]\(0^\circ < x < 90^\circ\)[/tex], we can utilize one of the fundamental trigonometric identities.
The trigonometric identity we will use is:
[tex]\[
\sin(x) = \cos(90^\circ - x)
\][/tex]
This identity tells us that the sine of an angle is equal to the cosine of its complement. Therefore, if [tex]\(\sin(x) = \cos(32^\circ)\)[/tex], then we must have:
[tex]\[
x = 90^\circ - 32^\circ
\][/tex]
Let's perform the subtraction:
[tex]\[
x = 90^\circ - 32^\circ = 58^\circ
\][/tex]
So, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\(\sin(x) = \cos(32^\circ)\)[/tex] within the given interval [tex]\(0^\circ < x < 90^\circ\)[/tex] is:
[tex]\[
x = 58^\circ
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{58^\circ}
\][/tex]
