To solve the problem, follow these steps:
1. Understand the relationship between the angles in a right triangle:
- In a right triangle, the two non-right angles are complementary, meaning that their measures add up to 90 degrees.
2. Use the complementary angle property:
- If [tex]\(\angle X\)[/tex] and [tex]\(\angle Z\)[/tex] are complementary angles, then [tex]\(\cos(X) = \sin(Z)\)[/tex]. This is due to the fact that [tex]\(\cos(\theta) = \sin(90^\circ - \theta)\)[/tex], where [tex]\(\theta\)[/tex] is an angle in the triangle.
3. Given information:
- We know [tex]\(\cos(X) = \frac{9}{11}\)[/tex].
4. Apply the trigonometric identity:
- Since [tex]\(\cos(X) = \sin(Z)\)[/tex] and we are given [tex]\(\cos(X) = \frac{9}{11}\)[/tex], it follows directly that [tex]\(\sin(Z) = \frac{9}{11}\)[/tex].
Thus, the value of [tex]\(\sin(Z)\)[/tex] is [tex]\(\frac{9}{11}\)[/tex].
So, the correct answer is
D. [tex]\(\frac{9}{11}\)[/tex].
