https://file.aiquickdraw.com/mathbot/file/1725470541612rg6g254f.jpg

Let's work through the image together. The problem involves three right triangles that are similar, with specific angles and side lengths provided. Here's a breakdown of the tasks:

### (a) Find the ratio of the length of the side adjacent to the \(41.7^\circ\) angle to the hypotenuse for each triangle.

1. **Triangle ABC**
   - Given side lengths: AC (hypotenuse) = 7.54 cm, AB (adjacent to \(41.7^\circ\)) = 5.63 cm.
   - The ratio: \( \frac{AB}{AC} = \frac{5.63}{7.54} \).

2. **Triangle DEF**
   - Given side lengths: DF (hypotenuse) = 10.62 cm, DE (adjacent to \(41.7^\circ\)) = 7.98 cm.
   - The ratio: \( \frac{DE}{DF} = \frac{7.98}{10.62} \).

3. **Triangle XYZ**
   - Given side lengths: XZ (hypotenuse) = 23.78 cm, XY (adjacent to \(41.7^\circ\)) = 17.79 cm.
   - The ratio: \( \frac{XY}{XZ} = \frac{17.79}{23.78} \).

We will calculate these ratios.

### (b) Find \(\sin(41.7^\circ)\), \(\cos(41.7^\circ)\), and \(\tan(41.7^\circ)\).

We will use a calculator to determine these trigonometric values.

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

Let me calculate all these values and then summarize them for you.Here are the calculations based on the information provided:

### (a) Ratios of the side adjacent to the \(41.7^\circ\) angle to the hypotenuse:
1. **Triangle ABC**: \( \frac{AB}{AC} = 0.747 \)
2. **Triangle DEF**: \( \frac{DE}{DF} = 0.751 \)
3. **Triangle XYZ**: \( \frac{XY}{XZ} = 0.748 \)

### (b) Trigonometric values for \(41.7^\circ\):
- \( \sin(41.7^\circ) \approx 0.665 \)
- \( \cos(41.7^\circ) \approx 0.747 \)
- \( \tan(41.7^\circ) \approx 0.891 \)

### (c) Which trigonometric function gives the ratio of sides in part (a)?

The ratios found in part (a) are approximately equal to \( \cos(41.7^\circ) \). Therefore, the trigonometric function that gives this ratio is **cosine**.

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Would you like further details on any of the steps? 

Here are five questions related to this topic:
1. How can we use the tangent function to find the length of the opposite side in these triangles?
2. What is the relationship between the sine and cosine of complementary angles?
3. How do we apply the Pythagorean theorem to check the calculations of side lengths in these triangles?
4. Why do all three triangles have the same trigonometric ratios despite having different side lengths?
5. How would the ratios change if the angle was different?

**Tip:** Always double-check if your calculator is in degree mode when dealing with angles in degrees.

The problem involves finding the ratio of the length of the side adjacent to the 41.7° angle to the hypotenuse for three similar right triangles, calculating trigonometric values for 41.7°, and determining which trigonometric function corresponds to the calculated ratios.

Explore the process of finding trigonometric ratios in similar right triangles with a 41.7° angle. Learn to calculate cos(41.7°), sin(41.7°), and tan(41.7°), and understand how these ratios apply to different triangles. Suitable for students in Grades 9-11.

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