To solve for 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 a fundamental trigonometric identity. This identity states that the sine of an angle is equal to the cosine of its complementary angle, i.e.,
[tex]\[
\sin(x) = \cos(90^\circ - x)
\][/tex]
Given the equation:
[tex]\[
\sin(x) = \cos(32^\circ)
\][/tex]
We can set up an equation using the identity above:
[tex]\[
\sin(x) = \cos(32^\circ) = \cos(90^\circ - (90^\circ - 32^\circ)) = \cos(58^\circ)
\][/tex]
Thus, to satisfy the given equation [tex]\(\sin(x) = \cos(32^\circ)\)[/tex], [tex]\( x \)[/tex] must be the complement of [tex]\( 32^\circ \)[/tex]. This means we set the complement of [tex]\( x \)[/tex] to [tex]\( 32^\circ \)[/tex]:
[tex]\[
x = 90^\circ - 32^\circ = 58^\circ
\][/tex]
Therefore, the correct value of [tex]\( x \)[/tex] is:
[tex]\[
\boxed{58^\circ}
\][/tex]
