Certainly! Let's solve the given problem step-by-step.
In a right triangle [tex]\( \triangle ABC \)[/tex] with [tex]\( \angle C \)[/tex] being the right angle, the angles [tex]\( \angle A \)[/tex] and [tex]\( \angle B \)[/tex] are complementary. This means that [tex]\( \angle A + \angle B = 90^\circ \)[/tex].
Given:
[tex]\[ \sin A = \frac{8}{9} \][/tex]
We need to find [tex]\( \cos B \)[/tex].
1. Complementary angles in a right triangle have a special relationship with the sine and cosine functions:
[tex]\[ \sin A = \cos B \][/tex]
2. Therefore, since [tex]\( \angle A \)[/tex] and [tex]\( \angle B \)[/tex] are complementary angles:
[tex]\[ \cos B = \sin A \][/tex]
3. Substitute the given value:
[tex]\[ \cos B = \frac{8}{9} \][/tex]
Thus, the value of [tex]\( \cos B \)[/tex] is [tex]\( \frac{8}{9} \)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{\frac{8}{9}} \][/tex]
