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Let's break down the question and solve it step by step.

### Part (a) – Ratios of Sides for Each Triangle

We are asked to find the ratio of the side opposite the angle 50.7° to the hypotenuse for each triangle:

1. **For triangle ABC:**
   - Opposite side \( AB = 9.8 \, \text{m} \)
   - Hypotenuse \( AC = 15.47 \, \text{m} \)
   - Ratio: \( \frac{AB}{AC} = \frac{9.8}{15.47} \)

2. **For triangle DEF:**
   - Opposite side \( DE = 29.4 \, \text{m} \)
   - Hypotenuse \( DF = 46.41 \, \text{m} \)
   - Ratio: \( \frac{DE}{DF} = \frac{29.4}{46.41} \)

3. **For triangle XYZ:**
   - Opposite side \( XY = 68.6 \, \text{m} \)
   - Hypotenuse \( XZ = 108.29 \, \text{m} \)
   - Ratio: \( \frac{XY}{XZ} = \frac{68.6}{108.29} \)

Let's calculate the ratios to the nearest hundredth.

### Part (b) – Trigonometric Values

We are asked to find:
- \( \sin 50.7^\circ \)
- \( \cos 50.7^\circ \)
- \( \tan 50.7^\circ \)

These values can be computed using a calculator or reference tables.

### Part (c) – Identify the Trigonometric Function

The trigonometric function that corresponds to the ratio of the opposite side to the hypotenuse is **sine**. Therefore, the correct answer is **sine**.

Let me first compute the ratios for part (a) and the trigonometric values for part (b).### Part (a) – Ratios of Sides:

1. \( \frac{AB}{AC} = 0.63 \)
2. \( \frac{DE}{DF} = 0.63 \)
3. \( \frac{XY}{XZ} = 0.63 \)

### Part (b) – Trigonometric Values:

- \( \sin 50.7^\circ = 0.77 \)
- \( \cos 50.7^\circ = 0.63 \)
- \( \tan 50.7^\circ = 1.22 \)

### Part (c) – Identifying the Trigonometric Function:

The ratio in part (a) corresponds to the **sine** function because it compares the opposite side to the hypotenuse. Therefore, the correct answer is **sine**.

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Would you like further details or explanations on any of the steps? Here are five follow-up questions to deepen your understanding:

1. How can you use the sine function to find missing sides in right triangles?
2. What would change if the angle were different, say 60°?
3. Can you derive the cosine value directly from the sine using complementary angles?
4. How would you use tangent if only the adjacent side were known?
5. Why do the three triangles in part (a) have the same ratio?

**Tip:** Memorize the key trigonometric functions (sine, cosine, and tangent) and their side relationships for right triangles to solve problems faster.

For each triangle, find the ratio of the side opposite 50.7° to the length of the hypotenuse. Round your answers to the nearest hundredth. Also, use the ALEKS calculator to find sin(50.7°), cos(50.7°), and tan(50.7°), and determine which trigonometric function gives each ratio of sides.

Learn how to find the ratio of the side opposite a 50.7° angle to the hypotenuse for three right triangles. This exercise involves calculating sine, cosine, and tangent values and understanding trigonometric functions. Suitable for students in Grades 9-12.

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