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The problem requires us to find the area of triangle \( \triangle DEF \) using trigonometry. We are provided the following information:

- One angle \( \angle EFD = 63^\circ \),
- Two sides: \( DE = 13.4 \), \( EF = 15 \),
- The base of the triangle \( DF = 20 \) is noted, but we are told only part of it is relevant.

The area of the triangle can be computed using the formula:

\[
\text{Area} = \frac{1}{2} b h
\]

However, to use trigonometry here, we need to determine the height relative to a base. We can use the given angle and the fact that this is a non-right triangle.

We can break down this problem as follows:

1. First, find the height (perpendicular from \( D \) to \( EF \)).
   - Use trigonometric functions like sine or cosine with the known angle \( 63^\circ \) and side lengths.
2. Multiply the base (which may be part of the side lengths) by the height and apply the formula for the area.

Let me calculate the area step by step.The area of triangle \( \triangle DEF \) is approximately \( 89.55 \).

Would you like further clarification on any step, or have any other questions? Here are some related questions that could help you understand more:

1. How do we use sine or cosine in trigonometry to find unknown sides or heights?
2. Why is the formula for the area of a triangle \( \frac{1}{2} \times \text{base} \times \text{height} \)?
3. Can you explain how angles and sides in non-right triangles relate to trigonometric functions?
4. What is the Law of Sines and how could it be applied in this context?
5. How do you determine which trigonometric function (sine, cosine, or tangent) to use?

**Tip:** When dealing with triangles, always ensure whether it's a right-angled triangle before choosing trigonometric methods—sine, cosine, or tangent—depending on the known sides or angles.

Use trigonometry to find the area of the triangle. Round and do not include units or commas.

Find the area of a triangle using trigonometric functions based on side lengths and angles. This problem explores trigonometric concepts such as sine and cosine to calculate height and area, suitable for students in Grades 9-11.

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