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 basic trigonometric identity.
Here's a step-by-step solution:
1. Understand the given equation:
We need to find [tex]\( x \)[/tex] such that:
[tex]\[
\cos(x) = \sin(14^\circ)
\][/tex]
2. Use the co-function identity:
One of the fundamental identities in trigonometry is:
[tex]\[
\sin(\theta) = \cos(90^\circ - \theta)
\][/tex]
This identity tells us that the sine of an angle is equal to the cosine of its complement.
3. Apply the identity:
Given that [tex]\( \sin(14^\circ) \)[/tex] can be written in terms of cosine, we have:
[tex]\[
\sin(14^\circ) = \cos(90^\circ - 14^\circ)
\][/tex]
Calculate the complement of [tex]\( 14^\circ \)[/tex]:
[tex]\[
90^\circ - 14^\circ = 76^\circ
\][/tex]
Thus:
[tex]\[
\sin(14^\circ) = \cos(76^\circ)
\][/tex]
4. Substitute back into the original equation:
From the original equation [tex]\( \cos(x) = \sin(14^\circ) \)[/tex], we now have:
[tex]\[
\cos(x) = \cos(76^\circ)
\][/tex]
5. Compare the cosine values:
Since [tex]\( \cos(x) = \cos(76^\circ) \)[/tex] and both the angles are in the range [tex]\( 0^\circ < x < 90^\circ \)[/tex], it follows that:
[tex]\[
x = 76^\circ
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( \cos(x) = \sin(14^\circ) \)[/tex] in the given range is [tex]\( 76^\circ \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{76^\circ}
\][/tex]
