To find the value of [tex]\( x \)[/tex] for which [tex]\( \sin(x) = \cos(32^\circ) \)[/tex] given that [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can make use of a fundamental trigonometric identity that relates sine and cosine. Specifically, the identity states that:
[tex]\[ \sin(x) = \cos(90^\circ - x) \][/tex]
Given [tex]\(\sin(x) = \cos(32^\circ)\)[/tex], we can use the identity to equate [tex]\( \sin(x) \)[/tex] with [tex]\( \cos(32^\circ) \)[/tex] as follows:
[tex]\[ \sin(x) = \cos(90^\circ - x) \][/tex]
This means we can set [tex]\(\cos(32^\circ)\)[/tex] equal to [tex]\(\cos(90^\circ - x)\)[/tex]. Therefore:
[tex]\[ \cos(32^\circ) = \cos(90^\circ - x) \][/tex]
In order for this equality to hold true, the arguments of the cosines must be equal, since the cosine function is one-to-one in the interval from [tex]\(0^\circ\)[/tex] to [tex]\(90^\circ\)[/tex]. Thus:
[tex]\[ 32^\circ = 90^\circ - x \][/tex]
To solve for [tex]\( x \)[/tex], we rearrange the equation:
[tex]\[ x = 90^\circ - 32^\circ \][/tex]
[tex]\[ x = 58^\circ \][/tex]
Therefore, the value of [tex]\( x \)[/tex] that satisfies [tex]\( \sin(x) = \cos(32^\circ) \)[/tex] in the interval [tex]\( 0^\circ < x < 90^\circ \)[/tex] is:
[tex]\[ \boxed{58^\circ} \][/tex]
Hence, the correct answer is:
A. [tex]\( 58^\circ \)[/tex]
