To find the value of [tex]\( x \)[/tex] for which [tex]\(\cos(x) = \sin(14^\circ) \)[/tex] within the range [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can use a fundamental trigonometric identity.
The identity states that:
[tex]\[
\sin(\theta) = \cos(90^\circ - \theta)
\][/tex]
for any angle [tex]\(\theta\)[/tex].
Given [tex]\(\sin(14^\circ)\)[/tex], we need [tex]\(\cos(x)\)[/tex] to be equal to [tex]\(\sin(14^\circ)\)[/tex]. Using the identity, we can write:
[tex]\[
\sin(14^\circ) = \cos(90^\circ - 14^\circ)
\][/tex]
Thus, if [tex]\(\cos(x) = \sin(14^\circ)\)[/tex], we must have:
[tex]\[
x = 90^\circ - 14^\circ
\][/tex]
So, calculating this gives:
[tex]\[
x = 90^\circ - 14^\circ = 76^\circ
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] for which [tex]\(\cos(x) = \sin(14^\circ) \)[/tex] is [tex]\( 76^\circ \)[/tex]. Hence, the correct answer is:
[tex]\[
\boxed{76^\circ}
\][/tex]
