To determine the value of [tex]\( x \)[/tex] such that [tex]\( \sin(x) = \cos(32^\circ) \)[/tex] for [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can use the trigonometric identity that relates sine and cosine:
[tex]\[
\sin(x) = \cos(90^\circ - x)
\][/tex]
From the given problem, we have:
[tex]\[
\sin(x) = \cos(32^\circ)
\][/tex]
Using the identity, we can rewrite [tex]\(\cos(32^\circ)\)[/tex]:
[tex]\[
\sin(x) = \cos(90^\circ - x)
\][/tex]
Thus, the equation becomes:
[tex]\[
\cos(90^\circ - x) = \cos(32^\circ)
\][/tex]
For these two cosine values to be equal, their arguments must be equal (since cosine is a periodic function, and we are considering angles in the range [tex]\(0^\circ < x < 90^\circ\)[/tex] where it is uniquely one-to-one):
[tex]\[
90^\circ - x = 32^\circ
\][/tex]
To find [tex]\( x \)[/tex], we solve for [tex]\( x \)[/tex] in the equation:
[tex]\[
90^\circ - x = 32^\circ
\][/tex]
Subtract [tex]\( 32^\circ \)[/tex] from both sides:
[tex]\[
90^\circ - 32^\circ = x
\][/tex]
So:
[tex]\[
x = 58^\circ
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\(\sin(x) = \cos(32^\circ)\)[/tex] is:
[tex]\[
\boxed{58^\circ}
\][/tex]
