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We are given a right triangle \( \triangle EFG \), where \( EF = 4.6 \), \( FG = 3 \), and we are asked to find which equation could be used to calculate the measure of \( \angle E \).

To determine the correct trigonometric equation, we need to identify which sides correspond to the angle \( \angle E \). In a right triangle:

- \( EF \) is the hypotenuse (longest side).
- \( FG \) is the side adjacent to \( \angle E \) (next to the angle).
- \( EG \) is the opposite side to \( \angle E \) (across from the angle).

### Identifying the correct trigonometric ratio:
Since we know the adjacent side \( FG = 3 \) and the hypotenuse \( EF = 4.6 \), the **cosine** ratio is the appropriate choice, as cosine is defined as:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Thus, to find \( m \angle E \), the equation should be:
\[
m \angle E = \cos^{-1} \left( \frac{3}{4.6} \right)
\]

### Conclusion:
The correct equation from the provided choices is:
\[
m \angle E = \cos^{-1} \left( \frac{3}{4.6} \right)
\]

Let me know if you need further details or explanations.

Here are 5 related questions to deepen understanding:
1. How can we use the sine ratio to find \( \angle E \)?
2. How would the equation change if we were finding \( \angle F \)?
3. What is the value of \( \angle E \) to the nearest degree?
4. How do we determine the length of the side \( EG \) using the Pythagorean theorem?
5. What other real-world applications use trigonometric ratios?

**Tip:** Always make sure to label the sides of a triangle correctly relative to the angle you're working with to choose the right trigonometric ratio.

Which equation could be used to find m∠E in ∆EFG?

This problem explores how to calculate angle E in triangle EFG using inverse trigonometric functions. With side lengths EF = 4.6 and FG = 3, we apply the cosine ratio to find the correct equation for solving the angle. Suitable for students in Grades 9-12.

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