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Qual é o valor da coordenada de 
\[y\] do ponto 
\[A\]?
Escolha 1 resposta:
Escolha 1 resposta:
(Escolha A)   
\[\operatorname{sen}(0)\]
A
\[\operatorname{sen}(0)\]
(Escolha B)   
\[\cos(0)\]
B
\[\cos(0)\]
(Escolha C)   
 
\[\operatorname{sen}\left(\dfrac{\pi}{2}\right)\]
C
 
\[\operatorname{sen}\left(\dfrac{\pi}{2}\right)\]
(Escolha D)   
 
\[\cos\left(\dfrac{\pi}{2}\right)\]
D
 
\[\cos\left(\dfrac{\pi}{2}\right)\]
(Escolha E)   
 
\[\operatorname{sen}\left(\dfrac{3\pi}{2}\right)\]
E
 
\[\operatorname{sen}\left(\dfrac{3\pi}{2}\right)\]
(Escolha F)   
 
\[\cos\left(\dfrac{3\pi}{2}\right)\]
F
 
\[\cos\left(\dfrac{3\pi}{2}\right)\]

To determine the \( y \)-coordinate of point \( A \), we can analyze the triangle using trigonometric principles, specifically the sine and cosine functions. Let's break down the steps.

### Given Information:
- \( BC = 21 \) (base)
- \( CA = 20 \) (height)
- \( AB = 29 \) (hypotenuse)
- Angle \( \alpha \) at point \( A \) is formed by sides \( CA \) and \( AB \).
- Point \( C \) has a right angle.

### Steps to Solve:

1. **Determine the Angle \( \alpha \)**: 
   The \( y \)-coordinate of \( A \) is equivalent to the sine of angle \( \alpha \) since \( C \) is a right angle and lies directly below \( A \).

2. **Use Sine Function**: 
   \( \sin(\alpha) = \frac{\text{opposite side}}{\text{hypotenuse}} \)
   In this triangle, the opposite side to angle \( \alpha \) is the length \( BC = 21 \) (as point \( C \) lies directly under \( A \)) and the hypotenuse is \( AB = 29 \).

   However, as the problem is asking for the \( y \)-coordinate specifically, this would be equivalent to the sine of \( \alpha \) or \( \sin(\alpha) \). Therefore:

   \[
   y = \sin(\alpha)
   \]

3. **Determine the Correct Trigonometric Identity**:
   Since the problem gives trigonometric choices, and based on the given options, the value that corresponds to \( \sin(\alpha) \) or the vertical height must be \( \sin\left(\frac{\pi}{2}\right) \), which equals 1.

### Conclusion:
The correct answer that corresponds to the \( y \)-coordinate of point \( A \) would be choice \( C \):

\[
\operatorname{sen}\left(\dfrac{\pi}{2}\right)
\]

This matches the height derived from the problem's given triangle, making it the correct option.

Do you want further details or have any questions? Here are some related questions you might find useful:

1. What is the significance of the sine and cosine functions in determining coordinates on a triangle?
2. How do we use the Pythagorean theorem in verifying the dimensions of a triangle?
3. Can we derive the other angles in the triangle using trigonometric functions?
4. How does changing the angle \( \alpha \) affect the coordinates of point \( A \)?
5. What would be the \( x \)-coordinate of point \( A \) based on the given triangle?

**Tip**: Always double-check the given options in multiple-choice questions by verifying with basic trigonometric identities!

Learn how to determine the y-coordinate of Point A using trigonometric functions. This problem involves applying the sine function to find the correct coordinate in a given triangle setup. Suitable for high school students studying trigonometry.

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