To determine the value of [tex]\( x \)[/tex] for which [tex]\( \sin(x) = \cos(32^\circ) \)[/tex] within the interval [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can use a fundamental trigonometric identity.
The identity we use here is:
[tex]\[ \sin(x) = \cos(90^\circ - x) \][/tex]
This relationship is useful because it tells us how the sine and cosine functions relate to angles that are complementary (sum up to [tex]\( 90^\circ \)[/tex]). Given the equation [tex]\( \sin(x) = \cos(32^\circ) \)[/tex], we can equate this to the identity:
[tex]\[ \sin(x) = \cos(32^\circ) \][/tex]
According to the identity, [tex]\( \cos(32^\circ) \)[/tex] can be rewritten using the sine function:
[tex]\[ \cos(32^\circ) = \sin(90^\circ - 32^\circ) \][/tex]
Therefore, we replace [tex]\( \cos(32^\circ) \)[/tex] with [tex]\( \sin(90^\circ - 32^\circ) \)[/tex] in our equation:
[tex]\[ \sin(x) = \sin(90^\circ - 32^\circ) \][/tex]
Since the sine function is one-to-one for [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can equate the arguments of the sine functions:
[tex]\[ x = 90^\circ - 32^\circ \][/tex]
Then we perform the subtraction:
[tex]\[ x = 58^\circ \][/tex]
Thus, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( \sin(x) = \cos(32^\circ) \)[/tex] in the given interval is:
[tex]\[ \boxed{58^\circ} \][/tex]
