https://r2.erweima.ai/mathbot/image/1726003959073t46cy55h.jpg

The given image shows a right triangle \( \triangle ABC \), where:

- \( \angle C = 90^\circ \),
- \( BC = 12 \),
- \( AC = 35 \),
- \( AB = 37 \), and
- \( \alpha \) is the angle at \( A \).

You are asked to find \( \sin(\alpha) \).

### Step-by-step Solution:

Since this is a right triangle, we can use the definition of sine for angle \( \alpha \):

\[
\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

In this case:

- The side opposite to \( \alpha \) is \( BC = 12 \),
- The hypotenuse is \( AB = 37 \).

Thus,

\[
\sin(\alpha) = \frac{12}{37}
\]

This is the exact value of \( \sin(\alpha) \).

Let me know if you need further details or clarifications!

Here are 5 related questions:
1. What is the value of \( \cos(\alpha) \) in this triangle?
2. How would you find \( \tan(\alpha) \)?
3. Can you find the length of \( AB \) using the Pythagorean theorem?
4. What is the value of \( \sin(\beta) \), where \( \beta \) is the angle at \( B \)?
5. What are the angles of the triangle?

**Tip**: In a right triangle, sine and cosine relate to the ratio of the sides, while the tangent gives the ratio of the opposite side to the adjacent side.

Solve for \sin(\alpha) in the right triangle with hypotenuse 37 and opposite side 12. Using the sine formula for right triangles, this solution walks through the steps with clear explanations. Suitable for students in Grades 9-11.

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