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The given problem shows a right triangle \( \triangle DEF \) with the lengths of sides \( DF = 7 \), \( DE = 9 \), and \( \angle E = 90^\circ \). We are tasked with determining the approximate value of \( \theta \), the angle opposite to side \( EF \).

To solve for \( \theta \), we can use trigonometric ratios. Specifically, since we know the lengths of the adjacent side \( DF \) and the hypotenuse \( DE \), we use the cosine function:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{7}{9}
\]

Now, calculate \( \theta \) by taking the inverse cosine (arccos):

\[
\theta = \cos^{-1}\left( \frac{7}{9} \right)
\]

Let me calculate this for you.The approximate value of \( \theta \) is \( 38.9^\circ \).

Thus, the correct answer is **b) 38.9°**.

Would you like more details on this calculation or have any other questions? Here are five related questions to expand on this topic:

1. How would you solve for \( \theta \) using the sine or tangent functions in this case?
2. What would happen if the triangle had different side lengths? How would that affect \( \theta \)?
3. How can we verify that this triangle satisfies the Pythagorean theorem?
4. What are the other trigonometric ratios (sine and tangent) for this triangle?
5. What is the value of the other acute angle in the triangle?

**Tip**: When solving right triangles, always check which trigonometric ratio to use based on the sides given relative to the angle.

Determine the approximate value of θ in the right triangle with sides DF = 7, DE = 9, and angle E = 90°.

Learn how to solve for angle θ in a right triangle using the cosine function. The problem involves sides DF = 7, DE = 9, and the angle E = 90°. This tutorial is suitable for students in Grades 9-12, covering trigonometry and the inverse cosine function.

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