To determine 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 well-known trigonometric identity.
The trigonometric identity we will use is:
[tex]\[ \cos(x) = \sin(90^\circ - x) \][/tex]
We are given:
[tex]\[ \cos(x) = \sin(14^\circ) \][/tex]
By the trigonometric identity, we can express:
[tex]\[ \sin(14^\circ) = \cos(x) = \sin(90^\circ - x) \][/tex]
To find [tex]\( x \)[/tex], we then need to solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ 14^\circ = 90^\circ - x \][/tex]
Let's solve for [tex]\( x \)[/tex]:
[tex]\[ 14^\circ + x = 90^\circ \][/tex]
[tex]\[ x = 90^\circ - 14^\circ \][/tex]
[tex]\[ x = 76^\circ \][/tex]
Thus, the value of [tex]\( x \)[/tex] that satisfies [tex]\( \cos(x) = \sin(14^\circ) \)[/tex] is:
[tex]\[ x = 76^\circ \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{76^\circ} \][/tex]
