To solve for the value of [tex]\( x \)[/tex] that satisfies [tex]\( \cos(x) = \sin(14^\circ) \)[/tex] within the range [tex]\( 0^\circ < x < 90^\circ \)[/tex], let's use a known trigonometric identity.
The identity states that:
[tex]\[ \cos(x) = \sin(90^\circ - x) \][/tex]
We need [tex]\( \cos(x) \)[/tex] to be equal to [tex]\( \sin(14^\circ) \)[/tex]. From the identity above, it follows that:
[tex]\[ \cos(x) = \sin(90^\circ - x) \][/tex]
For [tex]\( \cos(x) \)[/tex] to equal [tex]\( \sin(14^\circ) \)[/tex], we set:
[tex]\[ \sin(90^\circ - x) = \sin(14^\circ) \][/tex]
Since the sine function is periodic and symmetrical, the only way for this to hold true given the constraints [tex]\( 0^\circ < x < 90^\circ \)[/tex] is if:
[tex]\[ 90^\circ - x = 14^\circ \][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[ 90^\circ - x = 14^\circ \][/tex]
[tex]\[ 90^\circ - 14^\circ = x \][/tex]
[tex]\[ x = 76^\circ \][/tex]
Hence, the value of [tex]\( x \)[/tex] that satisfies [tex]\( \cos(x) = \sin(14^\circ) \)[/tex] is [tex]\( 76^\circ \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{76^\circ} \][/tex]
Indeed, checking the intermediate value:
The sine of [tex]\( 14^\circ \)[/tex] approximately equals [tex]\( 0.24192189559966773 \)[/tex], and thus [tex]\(\cos(76^\circ)\)[/tex] also matches this value, confirming our solution.
