To solve this problem, let's first recall that in a right triangle, the two non-right angles are complementary. This means that if one of those angles is \( \angle X \), the other angle, \( \angle Z \), satisfies the equation:
[tex]\[ \angle X + \angle Z = 90^\circ \][/tex]
Given this complementary relationship, we know that:
[tex]\[ \sin(Z) = \cos(X) \][/tex]
Now, we are given that:
[tex]\[ \cos(X) = \frac{9}{11} \][/tex]
Since \( \sin(Z) \) is equal to \( \cos(X) \) for complementary angles, we can directly write:
[tex]\[ \sin(Z) = \frac{9}{11} \][/tex]
Therefore, the correct value of \( \sin(Z) \) is:
[tex]\[ \boxed{\frac{9}{11}} \][/tex]
The correct choice is A.
