Let's solve the problem step-by-step to find the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\(\cos (x) = \sin (14^\circ)\)[/tex], given that [tex]\( 0^\circ < x < 90^\circ \)[/tex].
1. Recall a Key Trigonometric Identity:
One of the basic trigonometric identities is:
[tex]\[
\sin(90^\circ - \theta) = \cos(\theta)
\][/tex]
This identity tells us that the sine of an angle is equal to the cosine of its complementary angle.
2. Apply the Identity:
In our given problem, we need to find [tex]\( x \)[/tex] such that:
[tex]\[
\cos(x) = \sin(14^\circ)
\][/tex]
Using the trigonometric identity mentioned above, we rewrite:
[tex]\[
\sin(14^\circ) = \cos(76^\circ) \quad \text{since} \quad 14^\circ + 76^\circ = 90^\circ
\][/tex]
3. Compare the Equations:
From the problem, we have:
[tex]\[
\cos(x) = \sin(14^\circ)
\][/tex]
And by using the identity, we find that:
[tex]\[
\sin(14^\circ) = \cos(76^\circ)
\][/tex]
Therefore:
[tex]\[
\cos(x) = \cos(76^\circ)
\][/tex]
4. Determine [tex]\( x \)[/tex]:
Since the cosine function [tex]\(\cos(\theta)\)[/tex] is one-to-one and continuous in the interval [tex]\((0^\circ, 90^\circ)\)[/tex], equating the arguments gives:
[tex]\[
x = 76^\circ
\][/tex]
This is within the given range [tex]\(0^\circ < x < 90^\circ\)[/tex].
5. Select the Correct Answer:
Comparing this result with the provided choices:
- [tex]\( \text{A. } 76^\circ \)[/tex]
- [tex]\( \text{B. } 28^\circ \)[/tex]
- [tex]\( \text{C. } 31^\circ \)[/tex]
- [tex]\( \text{D. } 14^\circ \)[/tex]
The correct answer is:
[tex]\[
\boxed{76^\circ}
\][/tex]
