To determine [tex]\( \cos B \)[/tex] in a right triangle [tex]\( ABC \)[/tex], where [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] are complementary angles (meaning their sum is [tex]\(90^\circ\)[/tex]), and given that [tex]\( \sin A = \frac{8}{9} \)[/tex], follow these steps:
1. Understand Complementary Angles: Since [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] are complementary, [tex]\(\angle A + \angle B = 90^\circ\)[/tex]. This relationship implies that [tex]\(\cos B = \sin A\)[/tex]. This is a crucial property in trigonometry for complementary angles that states: in a right triangle, the sine of one angle is equal to the cosine of the other angle.
2. Given Information: We know from the problem that [tex]\( \sin A = \frac{8}{9} \)[/tex]. Using the complementary angle property:
[tex]\[
\cos B = \sin A
\][/tex]
3. Substitute the Known Value: Replace [tex]\(\sin A\)[/tex] with the given value:
[tex]\[
\cos B = \frac{8}{9}
\][/tex]
Therefore, by substituting the given values and using the properties of complementary angles in a right triangle, we conclude that:
[tex]\[
\cos B = \frac{8}{9}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\frac{8}{9}}
\][/tex]
Hence, the correct option is:
[tex]\[
\boxed{B}
\][/tex]
