To solve this problem, let's first recap some important trigonometric principles involving complementary angles. In a right triangle, the sum of the two non-right angles is [tex]\(90^\circ\)[/tex]. Two angles that sum to [tex]\(90^\circ\)[/tex] are called complementary angles.
Given:
- [tex]\(\angle X\)[/tex] and [tex]\(\angle Z\)[/tex] are complementary.
- [tex]\(\cos(X) = \frac{9}{11}\)[/tex].
Here's a step-by-step solution to find [tex]\(\sin(Z)\)[/tex]:
1. Understanding the Complementary Relationship:
- Since [tex]\(\angle X\)[/tex] and [tex]\(\angle Z\)[/tex] are complementary, we have:
[tex]\[
X + Z = 90^\circ
\][/tex]
- The sine of an angle is equal to the cosine of its complementary angle:
[tex]\[
\sin(Z) = \cos(X)
\][/tex]
2. Substitute the Given Value:
- We know [tex]\(\cos(X) = \frac{9}{11}\)[/tex].
- Therefore:
[tex]\[
\sin(Z) = \cos(X) = \frac{9}{11}
\][/tex]
Thus, the value of [tex]\(\sin(Z)\)[/tex] is [tex]\(\frac{9}{11}\)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{\frac{9}{11}} \][/tex]
From the given choices, the correct option is D.
