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The image depicts a right triangle \( \triangle ABC \) with the following measurements:

- \( \angle C = 90^\circ \) (a right angle).
- \( \overline{AC} = 5 \, \text{cm} \).
- \( \overline{AB} = 10 \, \text{cm} \).
- \( \angle A = 30^\circ \).
- \( \overline{BC} = b \) (the unknown side).

To solve for \( b \) (side \( BC \)), we can use trigonometry, specifically the sine or cosine functions, since angle \( A = 30^\circ \) and the sides adjacent or opposite to that angle are given.

1. **Using the sine function**:
   The sine of an angle in a right triangle is defined as:
   \[
   \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
   \]
   For \( \angle A = 30^\circ \), the side opposite is \( BC = b \), and the hypotenuse is \( AB = 10 \, \text{cm} \).
   \[
   \sin(30^\circ) = \frac{b}{10}
   \]
   Since \( \sin(30^\circ) = 0.5 \), we get:
   \[
   0.5 = \frac{b}{10}
   \]
   Solving for \( b \):
   \[
   b = 10 \times 0.5 = 5 \, \text{cm}
   \]

Thus, the correct equation to solve for \( b \) is:
\[
\sin(30^\circ) = \frac{b}{10}
\]

Would you like more details on trigonometry or help with other related concepts?

Here are 5 related questions to consider:
1. What is the cosine equation for this triangle?
2. How can we solve for the third side using the Pythagorean theorem?
3. What is the angle \( B \) in triangle \( \triangle ABC \)?
4. How can trigonometry be used to solve other right triangles?
5. What are some real-world applications of trigonometry in geometry?

**Tip**: In right triangles, always identify the given angle and its relationship to the known sides (opposite, adjacent, or hypotenuse) to choose the correct trigonometric function.

Which equation can be used to solve for b?

Learn how to solve for the unknown side 'b' in a right triangle using the sine function. Given a 30-degree angle and the lengths of adjacent sides, this example demonstrates key trigonometric principles and is suitable for students in Grades 9-10.

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