https://file.aiquickdraw.com/mathbot/file/1726392406524x60yyjir.jpg

In this image, we are working with a right triangle that has side lengths of 9, 40, and 41. Based on this information, the hypotenuse is 41, the leg opposite angle \(Y\) is 9, and the adjacent leg to angle \(Y\) is 40.

To calculate the trigonometric values \( \sin Y \), \( \cos Y \), and \( \tan Y \), we use the following definitions for a right triangle:

1. **Sine of \( Y \)**:  
   \[
   \sin Y = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{41}
   \]
   
2. **Cosine of \( Y \)**:  
   \[
   \cos Y = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{40}{41}
   \]
   
3. **Tangent of \( Y \)**:  
   \[
   \tan Y = \frac{\text{opposite}}{\text{adjacent}} = \frac{9}{40}
   \]

### Final values:
\[
\sin Y = \frac{9}{41}, \quad \cos Y = \frac{40}{41}, \quad \tan Y = \frac{9}{40}
\]

Would you like further details on how to calculate or use these values? Here are five related questions:

1. How are the trigonometric functions related to the Pythagorean theorem?
2. What is the exact value of angle \( Y \) using inverse trigonometric functions?
3. How can the values of \( \sin Y \), \( \cos Y \), and \( \tan Y \) help in solving real-world problems?
4. What are the reciprocal trigonometric functions for this triangle?
5. How do trigonometric identities apply to this right triangle?

**Tip:** When solving for trigonometric functions in right triangles, always remember the mnemonic "SOH-CAH-TOA" to help recall the relationships.

A right triangle has side lengths 9, 40, and 41 as shown below. Use these lengths to find sin Y, tan Y, and cos Y.

Discover how to calculate the trigonometric values sin Y, cos Y, and tan Y using the side lengths of a right triangle with sides 9, 40, and 41. This problem covers essential trigonometry concepts suitable for high school students in Grades 9-12.

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