To determine the value of [tex]\( x \)[/tex] for which [tex]\(\sin x = \cos 19^{\circ}\)[/tex], where [tex]\(0^{\circ} < x < 90^{\circ}\)[/tex], we can use a fundamental trigonometric identity that relates the sine and cosine functions:
[tex]\[ \sin x = \cos(90^{\circ} - x) \][/tex]
Given [tex]\(\sin x = \cos 19^{\circ}\)[/tex], we can set up the equation based on the identity:
[tex]\[ \sin x = \cos 19^{\circ} \][/tex]
Using the above identity, we can write:
[tex]\[ \sin x = \cos(90^{\circ} - 71^{\circ}) \][/tex]
For the values to match, we recognize that:
[tex]\[ x = 90^{\circ} - 19^{\circ} \][/tex]
Calculating this:
[tex]\[ x = 71^{\circ} \][/tex]
Therefore, the correct answer is:
B. [tex]\(71^{\circ}\)[/tex]
