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Find all sides and angles of the triangle

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

- \( \angle B = 62^\circ \)
- \( \angle C = 90^\circ \)
- \( AC = 22.6 \) cm

### Step 1: Find \( \angle A \)
Since \( \triangle ABC \) is a right triangle, the sum of the angles in a triangle is \( 180^\circ \). Therefore, \( \angle A \) can be found as:

\[
\angle A = 180^\circ - \angle B - \angle C = 180^\circ - 62^\circ - 90^\circ = 28^\circ
\]

### Step 2: Use trigonometric ratios to find \( AB \) and \( BC \)

1. **Finding \( AB \) (Hypotenuse):**

Using the sine function:
\[
\sin(62^\circ) = \frac{AC}{AB}
\]
\[
AB = \frac{AC}{\sin(62^\circ)} = \frac{22.6 \text{ cm}}{\sin(62^\circ)}
\]

Calculating:
\[
\sin(62^\circ) \approx 0.8829
\]
\[
AB = \frac{22.6 \text{ cm}}{0.8829} \approx 25.6 \text{ cm}
\]

2. **Finding \( BC \) (Adjacent side to \( \angle B \)):**

Using the tangent function:
\[
\tan(62^\circ) = \frac{AC}{BC}
\]
\[
BC = \frac{AC}{\tan(62^\circ)} = \frac{22.6 \text{ cm}}{\tan(62^\circ)}
\]

Calculating:
\[
\tan(62^\circ) \approx 1.8807
\]
\[
BC = \frac{22.6 \text{ cm}}{1.8807} \approx 12.0 \text{ cm}
\]

### Final Results:
- \( \angle A = 28^\circ \)
- \( AB \approx 25.6 \) cm
- \( BC \approx 12.0 \) cm

Would you like more details or have any questions?

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Here are 5 related questions for further exploration:

1. How would the triangle's sides change if \( \angle B \) were larger?
2. What is the area of \( \triangle ABC \) based on the calculated sides?
3. Can you use the Pythagorean theorem to verify the side lengths of \( \triangle ABC \)?
4. How would you find the length of side \( AB \) using the cosine of \( \angle B \)?
5. What happens to the trigonometric ratios if \( \angle A \) were given instead of \( \angle B \)?

**Tip:** When working with right triangles, always ensure the angle you reference corresponds to the correct side (opposite, adjacent, or hypotenuse) to use the right trigonometric function.

Discover how to find all sides and angles of a triangle with a detailed step-by-step solution. This problem involves a right triangle with known angles and side lengths, applying trigonometric ratios like sine and tangent functions. Suitable for high school students studying geometry and trigonometry.

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