https://r2.erweima.ai/mathbot/image/1726742237431qa4dzzj8.jpg

The image asks you to group numbers based on their sine values. We have the following numbers (in multiples of \( \pi \)):

1. \( 17.5\pi \)
2. \( -4.5\pi \)
3. \( -3.5\pi \)
4. \( 4.5\pi \)
5. \( 17\pi \)
6. \( 15\pi \)
7. \( -0.5\pi \)
8. \( 12.5\pi \)
9. \( -4\pi \)

The sine function is periodic with a period of \( 2\pi \), meaning that values of \( \theta \) that differ by multiples of \( 2\pi \) will have the same sine value. We can reduce these angles modulo \( 2\pi \) to find equivalent angles between \( 0 \) and \( 2\pi \).

Let’s reduce these values:

- \( 17.5\pi = 17.5\pi - 8(2\pi) = 1.5\pi \)
- \( -4.5\pi = -4.5\pi + 3(2\pi) = 1.5\pi \)
- \( -3.5\pi = -3.5\pi + 2(2\pi) = 0.5\pi \)
- \( 4.5\pi = 4.5\pi - 2(2\pi) = 0.5\pi \)
- \( 17\pi = 17\pi - 8(2\pi) = \pi \)
- \( 15\pi = 15\pi - 7(2\pi) = \pi \)
- \( -0.5\pi = -0.5\pi + (2\pi) = 1.5\pi \)
- \( 12.5\pi = 12.5\pi - 6(2\pi) = 0.5\pi \)
- \( -4\pi = -4\pi + 2(2\pi) = 0\pi \)

Now group the numbers that share the same sine value:
1. \( 1.5\pi \) (which corresponds to \( \sin(1.5\pi) = -1 \)): \( 17.5\pi \), \( -4.5\pi \), \( -0.5\pi \)
2. \( 0.5\pi \) (which corresponds to \( \sin(0.5\pi) = 1 \)): \( -3.5\pi \), \( 4.5\pi \), \( 12.5\pi \)
3. \( \pi \) (which corresponds to \( \sin(\pi) = 0 \)): \( 17\pi \), \( 15\pi \), \( -4\pi \)

Thus, the correct grouping is:

- Group 1: \( 17.5\pi \), \( -4.5\pi \), \( -0.5\pi \)
- Group 2: \( -3.5\pi \), \( 4.5\pi \), \( 12.5\pi \)
- Group 3: \( 17\pi \), \( 15\pi \), \( -4\pi \)

Would you like more details or have any further questions?

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Here are five related questions:
1. What is the period of the sine function, and why does it matter here?
2. How do we find equivalent angles when working with trigonometric functions?
3. What is the sine value of \( 0\pi \), \( \pi \), and \( 1.5\pi \)?
4. What is the geometric interpretation of these angles on the unit circle?
5. How does the cosine function behave with similar angles?

**Tip**: Always simplify angles by subtracting multiples of \( 2\pi \) to find equivalent angles within one full circle.

Jeweils drei Zahlen haben denselben Sinuswert. Gruppieren Sie. (Numbers: 17.5π, -4.5π, -3.5π, 4.5π, 17π, 15π, -0.5π, 12.5π, -4π)

Learn how to group numbers like 17.5π, -4.5π, and more by their sine values. Using the sine function's periodicity and modulo 2π, discover equivalent angles and how to group them. Suitable for students in Grades 9-12.

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