https://file.aiquickdraw.com/mathbot/file/1726771703188h59dk9qr.jpg

Let's work on solving the trigonometric ratios based on the triangle provided.

We have a right-angled triangle with the following sides:

- \( EG = 44 \) (adjacent to angle \( F \) and opposite to angle \( G \))
- \( FG = 55 \) (hypotenuse)
- \( EF = 33 \) (opposite to angle \( F \) and adjacent to angle \( G \))

### 1. **Cosine of angle \( F \):**
   \[
   \cos F = \frac{\text{Adjacent to } F}{\text{Hypotenuse}} = \frac{EG}{FG} = \frac{44}{55}
   \]
   Simplifying:
   \[
   \cos F = \frac{44}{55} = \frac{4}{5}
   \]

### 2. **Sine of angle \( G \):**
   \[
   \sin G = \frac{\text{Opposite to } G}{\text{Hypotenuse}} = \frac{EG}{FG} = \frac{44}{55}
   \]
   Again, simplifying:
   \[
   \sin G = \frac{44}{55} = \frac{4}{5}
   \]

### Final answers:
- \( \cos F = \frac{4}{5} \)
- \( \sin G = \frac{4}{5} \)

Let me know if you'd like further clarifications.

Here are 5 related questions for practice:
1. What is \( \sin F \) in this triangle?
2. What is \( \tan F \) in this triangle?
3. What is \( \cos G \)?
4. If \( \tan G \) is required, how would you compute it?
5. How can you confirm the Pythagorean theorem using the sides of this triangle?

**Tip**: For right triangles, the Pythagorean theorem is an essential tool for checking the side lengths.

Express the trigonometric ratios as fractions in simplest terms for the triangle shown with side lengths EG = 44, FG = 55, and EF = 33. Calculate cos F and sin G.

Learn how to calculate the cosine of angle F and sine of angle G in a right triangle with side lengths EG = 44, FG = 55, and EF = 33. This problem covers essential trigonometric concepts, including cosine and sine ratios, and uses the Pythagorean theorem to verify the side lengths. Suitable for students in Grades 9-11.

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