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 utilize a fundamental trigonometric identity that relates sine and cosine.
We know that [tex]\(\cos(\theta) = \sin(90^\circ - \theta)\)[/tex]. This is due to the complementary angle relationship in trigonometry where the sine of an angle is equal to the cosine of its complement.
Let's apply this identity to our problem:
1. Given [tex]\(\sin(x) = \cos(32^\circ)\)[/tex], we can rewrite [tex]\(\cos(32^\circ)\)[/tex] using the trigonometric identity:
[tex]\[\cos(32^\circ) = \sin(90^\circ - 32^\circ)\][/tex]
2. Substitute this back into the original equation:
[tex]\[\sin(x) = \sin(90^\circ - 32^\circ)\][/tex]
3. Simplify the right-hand side:
[tex]\[\sin(x) = \sin(58^\circ)\][/tex]
4. Since [tex]\(\sin(x) = \sin(58^\circ)\)[/tex] and knowing the range [tex]\(0^\circ < x < 90^\circ\)[/tex], we can conclude:
[tex]\[x = 58^\circ\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\(\boxed{58^\circ}\)[/tex].
