To find the value of [tex]\( x \)[/tex] for which [tex]\(\cos (x)=\sin \left(14^{\circ}\right)\)[/tex], given that [tex]\(0^{\circ}
The co-function identity states that:
[tex]\[
\cos(x) = \sin(90^\circ - x)
\][/tex]
Given [tex]\(\cos(x) = \sin(14^\circ)\)[/tex], we can set these equal and use the co-function identity to write:
[tex]\[
\sin(90^\circ - x) = \sin(14^\circ)
\][/tex]
Since the sine function is periodic and one-to-one in the interval [tex]\(0^\circ < x < 90^\circ\)[/tex], we equate the arguments:
[tex]\[
90^\circ - x = 14^\circ
\][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[
90^\circ - x = 14^\circ
\][/tex]
Subtract [tex]\(14^\circ\)[/tex] from both sides:
[tex]\[
90^\circ - 14^\circ = x
\][/tex]
Thus:
[tex]\[
x = 76^\circ
\][/tex]
So the value of [tex]\( x \)[/tex] is:
[tex]\[
\boxed{76^\circ}
\][/tex]
