To find the value of [tex]\( x \)[/tex] for which [tex]\(\cos(x) = \sin(14^\circ)\)[/tex] and [tex]\(0^\circ < x < 90^\circ\)[/tex], we can use a fundamental trigonometric identity.
Here are the steps to solve the problem:
1. Recall the Complementary Angle Identity:
One of the basic trigonometric identities is that the cosine of an angle is equal to the sine of its complement. Mathematically, this is expressed as:
[tex]\[
\cos(x) = \sin(90^\circ - x)
\][/tex]
2. Apply the Identity:
Given the equation [tex]\(\cos(x) = \sin(14^\circ)\)[/tex], we can use the above identity to rewrite the right-hand side in terms of cosine:
[tex]\[
\cos(x) = \sin(14^\circ) \quad \text{is equivalent to} \quad \cos(x) = \cos(90^\circ - 14^\circ)
\][/tex]
3. Find the Complementary Angle:
From our identity, we know:
[tex]\[
90^\circ - x = 14^\circ
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex], solve the equation step-by-step:
[tex]\[
x = 90^\circ - 14^\circ
\][/tex]
[tex]\[
x = 76^\circ
\][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( 76^\circ \)[/tex].
Therefore, the correct answer is:
D. [tex]\( 76^\circ \)[/tex]
