Answered

Explore a world of knowledge and get your questions answered on IDNLearn.com. Our Q&A platform offers reliable and thorough answers to ensure you have the information you need to succeed in any situation.

In the right triangle [tex]\(XYZ\)[/tex], [tex]\(\angle X\)[/tex] and [tex]\(\angle Z\)[/tex] are complementary angles, and [tex]\(\cos (\angle X)\)[/tex] is [tex]\(\frac{9}{11}\)[/tex]. What is [tex]\(\sin (\angle Z)\)[/tex]?

A. [tex]\(\frac{\sqrt{20}}{11}\)[/tex]

B. [tex]\(\frac{11}{9}\)[/tex]

C. [tex]\(\frac{\sqrt{20}}{9}\)[/tex]

D. [tex]\(\frac{9}{11}\)[/tex]

Sagot :

GinnyAnsweridn
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].
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Thank you for trusting IDNLearn.com with your questions. Visit us again for clear, concise, and accurate answers.