In a right triangle, the sum of the angles is always [tex]\(180^\circ\)[/tex]. Since we know that [tex]\(\angle C\)[/tex] is [tex]\(90^\circ\)[/tex], the remaining two angles, [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex], must be complementary, meaning their sum is [tex]\(90^\circ\)[/tex]. Therefore, we have [tex]\(\angle A + \angle B = 90^\circ\)[/tex].
The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the hypotenuse. Similarly, the cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse.
Given [tex]\(\sin A = \frac{8}{9}\)[/tex], we can use the property of complementary angles in right triangles, which states that [tex]\(\sin(\theta) = \cos(90^\circ - \theta)\)[/tex].
Therefore, [tex]\(\cos B\)[/tex] can be expressed as:
[tex]\[
\cos B = \sin A
\][/tex]
Given that [tex]\(\sin A = \frac{8}{9}\)[/tex], we substitute to find:
[tex]\[
\cos B = \frac{8}{9}
\][/tex]
Thus, the value of [tex]\(\cos B\)[/tex] is [tex]\(\frac{8}{9}\)[/tex], which corresponds to option B.
[tex]\[
\boxed{\frac{8}{9}}
\][/tex]
