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Let's go through each part of the problem:

### (a) Find the ratio of the length of the side opposite \( 58.8^\circ \) to the length of the hypotenuse for each triangle. Round your answers to the nearest hundredth.

1. **Triangle \( \triangle DEF \)**:
   - Opposite side: \( DE = 10.32 \) units
   - Hypotenuse: \( DF = 19.93 \) units
   - Ratio: \( \frac{DE}{DF} = \frac{10.32}{19.93} \)

2. **Triangle \( \triangle HIJ \)**:
   - Opposite side: \( HI = 41.82 \) units
   - Hypotenuse: \( HJ = 79.72 \) units
   - Ratio: \( \frac{HI}{HJ} = \frac{41.82}{79.72} \)

3. **Triangle \( \triangle PQR \)**:
   - Opposite side: \( PQ = 9.48 \) units
   - Hypotenuse: \( PR = 16.4 \) units
   - Ratio: \( \frac{PQ}{PR} = \frac{9.48}{16.4} \)

Let's calculate these ratios.

### (b) Use trigonometric functions to find \( \sin 58.8^\circ \), \( \cos 58.8^\circ \), and \( \tan 58.8^\circ \). Round your answers to the nearest hundredth.

Let's compute these values:

- \( \sin 58.8^\circ \)
- \( \cos 58.8^\circ \)
- \( \tan 58.8^\circ \)

### (c) Identify which trigonometric function gives each ratio of sides in part (a).

We will identify whether it's sine, cosine, or tangent.

Now, let's do the calculations:### Calculations:

**(a) Ratios of the side opposite \( 58.8^\circ \) to the hypotenuse:**
1. For \( \triangle DEF \): \( \frac{DE}{DF} = 0.52 \) (rounded to the nearest hundredth)
2. For \( \triangle HIJ \): \( \frac{HI}{HJ} = 0.52 \) (rounded to the nearest hundredth)
3. For \( \triangle PQR \): \( \frac{PQ}{PR} = 0.58 \) (rounded to the nearest hundredth)

**(b) Trigonometric function values for \( 58.8^\circ \):**
- \( \sin 58.8^\circ = 0.86 \)
- \( \cos 58.8^\circ = 0.52 \)
- \( \tan 58.8^\circ = 1.65 \)

**(c) Identification of the trigonometric function that matches the ratios:**
The ratios \( \frac{DE}{DF} \), \( \frac{HI}{HJ} \), and \( \frac{PQ}{PR} \) are closest to the value of \( \sin 58.8^\circ \), which suggests that these ratios represent the sine function.

**Answer for (c):** The correct trigonometric function is **sine**.

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Would you like further details or have any questions? 

### Related Questions:
1. How do you find the ratios using cosine and tangent functions?
2. How would the ratios change if the angles were different?
3. What is the relationship between the sine and cosine of complementary angles?
4. How do these values help in solving real-world problems involving right triangles?
5. What are some applications of trigonometric ratios in engineering?

**Tip:** Always check the units when calculating trigonometric values, especially when rounding.

For each triangle, find the ratio of the length of the side opposite to 58.8° to the length of the hypotenuse. Then, use the trigonometric functions to find sin 58.8°, cos 58.8°, and tan 58.8°. Finally, identify which trigonometric function gives each ratio of sides.

Explore how to calculate the ratios of sides in right triangles using sine, cosine, and tangent. This problem covers the calculation of trigonometric ratios for a 58.8° angle and is suitable for students in Grades 9-10.

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