△DEF has a right angle at E,  DE = 30, EF = 72, and DF = 78. Find cos D.

Given the right triangle \( \triangle DEF \) with a right angle at \( E \), and the side lengths \( DE = 30 \), \( EF = 72 \), and \( DF = 78 \), we are required to find \( \cos D \).

### Steps to Solve

1. **Use the definition of cosine**: 
   \[
   \cos D = \frac{\text{Adjacent side to } D}{\text{Hypotenuse}}
   \]
   In \( \triangle DEF \):
   - The hypotenuse is \( DF = 78 \).
   - The side adjacent to angle \( D \) is \( DE = 30 \).

2. **Calculate \( \cos D \)**:
   \[
   \cos D = \frac{DE}{DF} = \frac{30}{78}
   \]

3. **Simplify the fraction**:
   \[
   \cos D = \frac{30}{78} = \frac{5}{13}
   \]

So, the value of \( \cos D \) is \( \frac{5}{13} \).

Would you like any further details or have any other questions?

Here are some related questions you might find interesting:
1. How do you find \( \sin D \) in the same triangle?
2. What is the area of \( \triangle DEF \)?
3. How do you find the length of the altitude from \( E \) to \( DF \)?
4. Can you verify the result using the Pythagorean theorem?
5. What is \( \tan D \) in this triangle?

**Tip:** Remember that the cosine of an angle in a right triangle is always the ratio of the adjacent side to the hypotenuse.

Discover how to find cos D in triangle DEF with a right angle at E, and side lengths DE = 30, EF = 72, and DF = 78. Step-by-step solution using the definition of cosine. Suitable for high school students studying trigonometry.

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