To find the value of [tex]\( x \)[/tex] for which [tex]\(\cos(x) = \sin(14^\circ)\)[/tex] within the interval [tex]\(0^\circ < x < 90^\circ\)[/tex], we can use the complementary angle identity in trigonometry.
Recall the complementary angle identity:
[tex]\[ \sin(90^\circ - \theta) = \cos(\theta) \][/tex]
We can rewrite [tex]\(\cos(x)\)[/tex] by using the fact that:
[tex]\[ \cos(x) = \sin(90^\circ - x) \][/tex]
Given in the question:
[tex]\[ \cos(x) = \sin(14^\circ) \][/tex]
From the complementary angle identity, we can equate:
[tex]\[ \sin(90^\circ - x) = \sin(14^\circ) \][/tex]
For the sine function, one key property is that if [tex]\(\sin(A) = \sin(B)\)[/tex], then [tex]\( A \)[/tex] can be equal to [tex]\( B \)[/tex] or [tex]\( 180^\circ - B \)[/tex].
But since [tex]\(0^\circ < x < 90^\circ\)[/tex], it is sufficient to consider only the direct equality due to the range restriction:
[tex]\[ 90^\circ - x = 14^\circ \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ 90^\circ - 14^\circ = x \][/tex]
[tex]\[ x = 76^\circ \][/tex]
Thus, the value of [tex]\( x \)[/tex] is:
[tex]\[ \boxed{76^\circ} \][/tex]
