To solve the problem where we need to find the value of [tex]\( x \)[/tex] such that [tex]\(\sin(x) = \cos(32^\circ)\)[/tex] within the range [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can use the trigonometric identity:
[tex]\[ \sin(x) = \cos(90^\circ - x) \][/tex]
Given this identity, we set the argument of the cosine function on the right-hand side to be equal to 32 degrees:
[tex]\[ \cos(32^\circ) \][/tex]
Thus, we have:
[tex]\[ \sin(x) = \cos(32^\circ) \][/tex]
Using the identity:
[tex]\[ \sin(x) = \cos(90^\circ - x) \][/tex]
We can equate the right-hand sides:
[tex]\[ \cos(32^\circ) = \cos(90^\circ - x) \][/tex]
For the equality to hold true within the specified range, it must be that:
[tex]\[ 90^\circ - x = 32^\circ \][/tex]
To find [tex]\( x \)[/tex], solve the equation:
[tex]\[ 90^\circ - x = 32^\circ \][/tex]
Subtract 32 degrees from 90 degrees:
[tex]\[ 90^\circ - 32^\circ = x \][/tex]
Thus, we get:
[tex]\[ x = 58^\circ \][/tex]
Therefore, the value of [tex]\( x \)[/tex] that satisfies the condition [tex]\(\sin(x) = \cos(32^\circ)\)[/tex] within the range [tex]\( 0^\circ < x < 90^\circ \)[/tex] is:
[tex]\[ x = 58^\circ \][/tex]
So the correct answer is:
[tex]\[ \boxed{58^\circ} \][/tex]
