To find the value of [tex]\( x \)[/tex] that satisfies [tex]\(\sin(x) = \cos(32^\circ)\)[/tex] where [tex]\(0^\circ < x < 90^\circ\)[/tex], we can utilize trigonometric identities.
One of the fundamental trigonometric identities is that [tex]\(\sin(x) = \cos(90^\circ - x)\)[/tex].
Given this identity, we can rewrite [tex]\(\cos(32^\circ)\)[/tex] in a way that relates to [tex]\(\sin(x)\)[/tex]:
[tex]\[
\cos(32^\circ) = \sin(90^\circ - 32^\circ)
\][/tex]
Therefore, we need:
[tex]\[
\sin(x) = \sin(90^\circ - 32^\circ)
\][/tex]
From this equation, we can see that:
[tex]\[
x = 90^\circ - 32^\circ
\][/tex]
Now, compute the value:
[tex]\[
x = 58^\circ
\][/tex]
Thus, the value of [tex]\( x \)[/tex] that satisfies the given condition [tex]\(\sin(x) = \cos(32^\circ)\)[/tex] is [tex]\( \boxed{58^\circ} \)[/tex].
