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Given triangle *DEF*, if *e*, *d*, and *f* are known, which can be used to solve for Ð*F*?

Given triangle \( DEF \) with sides \( e \), \( d \), and \( f \), we can use the Law of Cosines to solve for the angle \( \angle F \).

The Law of Cosines states:
\[
e^2 = d^2 + f^2 - 2df \cdot \cos(\angle F)
\]

To solve for \( \angle F \), we rearrange the formula to isolate \( \cos(\angle F) \):
\[
\cos(\angle F) = \frac{d^2 + f^2 - e^2}{2df}
\]

Then, we find \( \angle F \) by taking the inverse cosine (arccos) of the result:
\[
\angle F = \cos^{-1} \left( \frac{d^2 + f^2 - e^2}{2df} \right)
\]

Thus, with the known values of \( e \), \( d \), and \( f \), you can determine the measure of \( \angle F \).

Would you like a detailed explanation or have any questions? Here are some further questions you might consider:

1. How do you use the Law of Cosines to solve for other angles in a triangle?
2. What are the properties of the Law of Sines and when is it used?
3. How do you verify the result obtained from the Law of Cosines?
4. Can you solve for \( \angle F \) using other trigonometric methods?
5. How does the Law of Cosines relate to the Pythagorean theorem?

### Tip:
When using the Law of Cosines, ensure all sides and angles are measured in the same units (degrees or radians) and double-check your calculations to avoid errors.

Learn how to solve for angle F in triangle DEF using the Law of Cosines. Understand the formula e^2 = d^2 + f^2 - 2df * cos(angle F) and the process of using inverse cosine to find angle F. Suitable for advanced high school students and anyone interested in trigonometry.

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