To find the value of [tex]\( x \)[/tex] such that [tex]\(\cos(x) = \sin(14^\circ)\)[/tex] for [tex]\(0^\circ < x < 90^\circ\)[/tex], we can use a trigonometric identity:
We know that [tex]\(\sin(\theta) = \cos(90^\circ - \theta)\)[/tex].
Given [tex]\(\cos(x) = \sin(14^\circ)\)[/tex], we can rewrite the right-hand side using the identity:
[tex]\[
\sin(14^\circ) = \cos(90^\circ - 14^\circ)
\][/tex]
Now we have:
[tex]\[
\cos(x) = \cos(90^\circ - 14^\circ)
\][/tex]
Since the cosine function is equal to itself if the angles are the same:
[tex]\[
x = 90^\circ - 14^\circ
\][/tex]
Simplifying the expression:
[tex]\[
x = 76^\circ
\][/tex]
So the value of [tex]\( x \)[/tex] is [tex]\(76^\circ\)[/tex].
Hence, the correct answer is:
A. [tex]\(76^{\circ}\)[/tex]
