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We are given a right triangle where \( a = 2 \, \text{cm} \), \( b = 5 \, \text{cm} \), and we need to find the sine, cosine, and tangent of angle \( \beta \).

1. **Step 1: Use Pythagoras' Theorem to find the hypotenuse** (\( c \)):
   \[
   c = \sqrt{a^2 + b^2} = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}
   \]
   So, \( c = \sqrt{29} \).

2. **Step 2: Calculate the trigonometric ratios for \( \beta \)**:
   - **Sine**: \( \sin \beta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} = \frac{2}{\sqrt{29}} \)
   - **Cosine**: \( \cos \beta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c} = \frac{5}{\sqrt{29}} \)
   - **Tangent**: \( \tan \beta = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} = \frac{2}{5} \)

3. **Step 3: Exact values**:
   - \( \sin \beta = \frac{2}{\sqrt{29}} \)
   - \( \cos \beta = \frac{5}{\sqrt{29}} \)
   - \( \tan \beta = \frac{2}{5} \)

4. **Step 4: Approximate values**:
   - \( \sin \beta \approx \frac{2}{\sqrt{29}} \approx 0.371 \)
   - \( \cos \beta \approx \frac{5}{\sqrt{29}} \approx 0.928 \)
   - \( \tan \beta = \frac{2}{5} = 0.4 \)

Thus, the completed table would be:

|              | Exact Values          | Approximate Values |
|--------------|-----------------------|--------------------|
| \( \sin \beta \) | \( \frac{2}{\sqrt{29}} \)  | \( 0.371 \)          |
| \( \cos \beta \) | \( \frac{5}{\sqrt{29}} \)  | \( 0.928 \)          |
| \( \tan \beta \) | \( \frac{2}{5} \)          | \( 0.4 \)            |

Would you like further details or clarification?

### Relative Questions:
1. How can the Pythagorean theorem be applied in non-right-angled triangles?
2. Why do we rationalize denominators like \( \frac{2}{\sqrt{29}} \) in some cases?
3. How are sine, cosine, and tangent related to real-world applications like engineering or physics?
4. What are other methods to find angle \( \beta \) besides using trigonometric ratios?
5. How can these concepts be applied to solve triangles with unknown sides?

### Tip:
Always double-check whether a triangle is right-angled before using the Pythagorean theorem, as it only applies to right triangles.

Find sine, cosine, and tangent of β, given that a = 2 cm and b = 5 cm. Present both the exact value and an approximate value (accurate up to three or more decimals).

Learn how to calculate the sine, cosine, and tangent of angle β in a right triangle with given sides a = 2 cm and b = 5 cm. This problem uses trigonometric ratios and the Pythagorean theorem, suitable for students in Grades 9-10.

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