To determine the value of [tex]\(x\)[/tex] for which [tex]\(\sin(x) = \cos(32^\circ)\)[/tex] within the range [tex]\(0^\circ < x < 90^\circ\)[/tex], we can use a fundamental trigonometric identity that relates sine and cosine.
Recall the cofunction identity, which states that:
[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. Here, we need [tex]\(\sin(x)\)[/tex] to equal [tex]\(\cos(32^\circ)\)[/tex]. Using the cofunction identity, we can express this as:
[tex]\[
\sin(x) = \cos(32^\circ) = \cos(90^\circ - y)
\][/tex]
For some angle [tex]\(y\)[/tex]. From the identity, we see that the cosine of an angle is equal to the sine of its complement. Therefore, we can set:
[tex]\[
32^\circ = 90^\circ - x
\][/tex]
To solve for [tex]\(x\)[/tex], isolate [tex]\(x\)[/tex] on one side of the equation:
[tex]\[
x = 90^\circ - 32^\circ
\][/tex]
Subtract the values:
[tex]\[
x = 58^\circ
\][/tex]
Thus, the value of [tex]\(x\)[/tex] for which [tex]\(\sin(x) = \cos(32^\circ)\)[/tex] in the specified range is:
[tex]\[
\boxed{58^\circ}
\][/tex]
