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Encontre 
\[\operatorname{tg}(\beta)\] no triângulo.
Escolha 1 resposta:
Escolha 1 resposta:
(Opção A, verificada)   
 
\[\dfrac{8}{15}\]
A
 
\[\dfrac{8}{15}\]
(Escolha B)   
 
\[\dfrac{15}{8}\]
B
 
\[\dfrac{15}{8}\]
(Escolha C)   
 
\[\dfrac{15}{17}\]
C
 
\[\dfrac{15}{17}\]
(Escolha D)   
 
\[\dfrac{8}{17}\]
D
 
\[\dfrac{8}{17}\] portugues

To solve the problem of finding \(\tan(\beta)\) in the given triangle, we can analyze the triangle using trigonometric principles.

### Step-by-Step Solution:

1. **Identify the sides of the triangle with respect to angle \(\beta\):**
   - **Opposite side:** The side opposite to angle \(\beta\) is \(8\) (side \(AC\)).
   - **Adjacent side:** The side adjacent to angle \(\beta\) is \(15\) (side \(BC\)).
   - **Hypotenuse:** The hypotenuse is \(17\) (side \(AB\)).

2. **Recall the tangent function:**
   \[
   \tan(\beta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AC}{BC}
   \]

3. **Calculate \(\tan(\beta)\):**
   \[
   \tan(\beta) = \frac{8}{15}
   \]

### Conclusion:
The correct answer is \(\boxed{\frac{8}{15}}\).

Would you like more details, or do you have any other questions?

### Follow-up Questions:
1. What is the cosine of angle \(\beta\) in the given triangle?
2. How would you calculate the sine of angle \(\beta\)?
3. What would be the value of \(\tan(\beta)\) if the sides were rearranged?
4. How can you verify that the triangle is a right triangle?
5. How does the Pythagorean theorem relate to this problem?

### Tip:
Remember that the tangent function is the ratio of the lengths of the opposite side to the adjacent side in a right triangle.

Learn how to find \( \tan(\beta) \) in a triangle where one angle is \( \beta \) using trigonometric principles. Step-by-step solution provided with sides 8, 15, and 17. Suitable for high school students learning trigonometry.

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