To solve the problem of finding the value of [tex]\( x \)[/tex] such that [tex]\(\cos(x) = \sin(14^\circ)\)[/tex] within the range [tex]\(0^\circ < x < 90^\circ\)[/tex], we can use a fundamental trigonometric identity.
### Step-by-Step Solution:
1. Understanding the Trigonometric Identity:
- One of the fundamental trigonometric identities states that for any angle [tex]\(\theta\)[/tex]:
[tex]\[
\cos(\theta) = \sin(90^\circ - \theta)
\][/tex]
2. Applying the Identity:
- In this problem, we need to find [tex]\( x \)[/tex] such that:
[tex]\[
\cos(x) = \sin(14^\circ)
\][/tex]
- According to the identity stated above:
[tex]\[
\sin(14^\circ) = \cos(76^\circ)
\][/tex]
- Therefore, we have:
[tex]\[
\cos(x) = \cos(76^\circ)
\][/tex]
3. Equating the Angles:
- Since the cosine function is equal when their angles are equal (as [tex]\(0^\circ < x < 90^\circ\)[/tex]):
[tex]\[
x = 76^\circ
\][/tex]
So, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\(\cos(x) = \sin(14^\circ)\)[/tex] within the given range is:
[tex]\[
\boxed{76^\circ}
\][/tex]
