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Let's analyze the given problem:

The problem involves a right triangle \( \triangle DEF \) with the right angle at \( E \). The question asks which expressions are equivalent to \( \cos(\angle E) \).

### Triangle \( \triangle DEF \)
- \( \angle D \) is the angle opposite the side \( EF \).
- \( \angle F \) is the angle opposite the side \( DE \).
- \( \angle E \) is the right angle, opposite the hypotenuse \( DF \).

### Understanding \( \cos(\angle E) \)
In a right triangle, the cosine of an angle is defined as:
\[ \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \]

For \( \cos(\angle E) \):
- The adjacent side to \( \angle E \) is \( DE \).
- The hypotenuse is \( DF \).

Thus:
\[ \cos(\angle E) = \frac{DE}{DF} \]

### Equivalent Expressions
Given the choices:
1. \( \frac{DP}{EF} \)
2. \( \frac{\text{length of side opposite to } \angle E}{\text{length of hypotenuse}} \)
3. \( \sin(\angle F) \)
4. \( \frac{DE}{DF} \)

We will evaluate each:
1. \( \frac{DP}{EF} \) - Not relevant; \( DP \) and \( EF \) do not correspond to adjacent and hypotenuse for \( \angle E \).
2. \( \frac{\text{length of side opposite to } \angle E}{\text{length of hypotenuse}} \) - This describes \( \sin(\angle E) \), not \( \cos(\angle E) \).
3. \( \sin(\angle F) \) - This is equivalent because \( \sin(\theta) \) for an angle in a right triangle is defined as \( \frac{\text{opposite side}}{\text{hypotenuse}} \). Since \( \angle F \) is complementary to \( \angle E \) (\( \angle F = 90^\circ - \angle E \)), \( \sin(\angle F) = \cos(\angle E) \).
4. \( \frac{DE}{DF} \) - This directly matches the definition of \( \cos(\angle E) \).

### Correct Answers
Therefore, the correct expressions equivalent to \( \cos(\angle E) \) are:
- \( \sin(\angle F) \)
- \( \frac{DE}{DF} \)

Would you like further details or have any questions?

Here are some related questions for further practice:
1. What is the sine of \( \angle D \) in \( \triangle DEF \)?
2. What is the cosine of \( \angle F \) in \( \triangle DEF \)?
3. How would you find \( \tan(\angle E

Explore the concept of cosine in a right triangle through equivalent expressions. Learn how cosine relates to adjacent and hypotenuse sides, and discover which expressions are equivalent to cos(angle E) in triangle DEF.

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