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In the given right triangle \( \triangle ABC \), we have:

1. The side \( BC = 4 \) (opposite to angle \( A \)),
2. The side \( CA = 9 \) (adjacent to angle \( A \)),
3. The hypotenuse \( AB \) (unknown).

The problem shows the calculation for finding angle \( B \) using the sine function:
\[
\sin(B) = \frac{4}{\sqrt{97}}
\]
Here, \( \sqrt{97} \) is likely the hypotenuse, calculated using the Pythagorean theorem.

### Calculating Hypotenuse \( AB \)
Using the Pythagorean theorem:
\[
AB = \sqrt{BC^2 + CA^2} = \sqrt{4^2 + 9^2} = \sqrt{16 + 81} = \sqrt{97}
\]
So, \( AB = \sqrt{97} \).

### Finding Angle \( B \)
Given:
\[
\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{\sqrt{97}}
\]
To find \( B \), use the inverse sine function:
\[
B = \sin^{-1} \left( \frac{4}{\sqrt{97}} \right)
\]

### Finding Angle \( A \)
Since \( \triangle ABC \) is a right triangle:
\[
A = 90^\circ - B
\]

Would you like the final calculated values for angles \( A \) and \( B \)?

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Here are some related questions to explore further:

1. How do we calculate the hypotenuse if the side lengths are different?
2. What other trigonometric ratios could we use to find angle \( A \)?
3. How can we verify that the triangle's angles sum up to \( 180^\circ \)?
4. What is the value of \( \cos(B) \) for this triangle?
5. How would these calculations change if this was not a right triangle?

**Tip:** Remember, for right triangles, knowing two sides allows you to find the hypotenuse and all angle measures easily with trigonometric functions.

I need help finding angle A in the right triangle where side BC is 4, side CA is 9, and angle B is calculated using sin(B) = 4 / sqrt(97).

Learn how to find angles in a right triangle with given side lengths, using trigonometric ratios and the Pythagorean theorem. In this example, we calculate angle B with sin(B) = 4 / sqrt(97) and find angle A by complementarity. Ideal for Grades 9-10 students studying trigonometry.

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