To solve the equation [tex]\(\sin x = \cos 19^\circ\)[/tex] where [tex]\(0^\circ < x < 90^\circ\)[/tex], we can use a fundamental trigonometric identity.
The identity states that for any angle [tex]\(\theta\)[/tex]:
[tex]\[
\sin(\theta) = \cos(90^\circ - \theta)
\][/tex]
Given [tex]\(\sin x = \cos 19^\circ\)[/tex], let's express [tex]\(\cos 19^\circ\)[/tex] in terms of sine:
[tex]\[
\cos 19^\circ = \sin (90^\circ - 19^\circ)
\][/tex]
Thus, the equation [tex]\(\sin x = \cos 19^\circ\)[/tex] becomes:
[tex]\[
\sin x = \sin (90^\circ - 19^\circ)
\][/tex]
Since [tex]\(\sin x = \sin (90^\circ - 19^\circ)\)[/tex], we can equate the angles:
[tex]\[
x = 90^\circ - 19^\circ
\][/tex]
Now, perform the calculation:
[tex]\[
x = 71^\circ
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(\sin x = \cos 19^\circ\)[/tex] is [tex]\(x = 71^\circ\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{71^\circ}
\][/tex]
Hence, the correct choice is B. 71 degrees.
