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.