https://r2.erweima.ai/mathbot/image/1724608546796f4hba9zf.jpg

Qual é o valor da coordenada de 
\[y\] do ponto 
\[A\]?
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The diagram shows a unit circle with an angle of \(\frac{\pi}{6}\) measured from the positive x-axis to point \(A\). The goal is to find the \(y\)-coordinate of point \(A\).

### Understanding the Problem:
1. **Unit Circle**: The circle has a radius of 1.
2. **Angle \(\frac{\pi}{6}\)**: In radians, \(\frac{\pi}{6}\) corresponds to \(30^\circ\).
3. **Coordinates of Point \(A\)**: On the unit circle, the coordinates of a point corresponding to an angle \(\theta\) are given by:
   \[
   (x, y) = (\cos(\theta), \sin(\theta))
   \]

### Calculation:
For \(\theta = \frac{\pi}{6}\), we calculate:
- \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\)
- Therefore, the \(y\)-coordinate of point \(A\) is \( \boxed{\frac{1}{2}} \).

### Answer:
The correct \(y\)-coordinate of point \(A\) is \(\boxed{\frac{1}{2}}\).

Would you like to explore this further, or do you have any other questions?

### Follow-up Questions:
1. What is the x-coordinate of point \(A\) in this diagram?
2. How does the radius of the circle affect the coordinates of point \(A\)?
3. What would the coordinates be if the angle were \(\frac{\pi}{4}\)?
4. How can you find the coordinates of point \(A\) using the Pythagorean identity?
5. What is the cosine of \(\frac{\pi}{6}\), and how is it related to this problem?

### Tip:
On the unit circle, the sine of an angle gives the y-coordinate, and the cosine gives the x-coordinate for any angle.

Discover how to find the y-coordinate of Point A on a unit circle at θ = π/6 radians using trigonometric principles. This problem explores the concept of coordinates on the unit circle and involves calculating the sine function for a specific angle. Suitable for advanced high school students in Grades 10-12.

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