To find the value of [tex]\( x \)[/tex] for which [tex]\( \cos(x) = \sin(14^\circ) \)[/tex] given that [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can use a property of complementary angles in trigonometry.
Here are the steps to solve the problem:
1. Recall the Complementary Angle Relationship: In trigonometry, the sine of an angle is equal to the cosine of its complementary angle. Specifically, for any angle [tex]\( \theta \)[/tex],
[tex]\[
\sin(90^\circ - \theta) = \cos(\theta)
\][/tex]
2. Given Equation: We are given that
[tex]\[
\cos(x) = \sin(14^\circ)
\][/tex]
3. Apply the Complementary Angle Property: According to the complementary angle property, we have
[tex]\[
\sin(14^\circ) = \cos(76^\circ)
\][/tex]
4. Comparison: This tells us that
[tex]\[
\cos(x) = \cos(76^\circ)
\][/tex]
5. Determine [tex]\( x \)[/tex]: Since cosine is a one-to-one function in the interval [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can conclude that
[tex]\[
x = 76^\circ
\][/tex]
Thus, the value of [tex]\( x \)[/tex] that satisfies [tex]\( \cos(x) = \sin(14^\circ) \)[/tex] in the interval [tex]\( 0^\circ < x < 90^\circ \)[/tex] is [tex]\( 76^\circ \)[/tex].
So, the answer is [tex]\( \boxed{76^\circ} \)[/tex].
