https://file.aiquickdraw.com/mathbot/file/17285334427118f3pv2o3.jpg

The given image shows a unit circle with an angle \( t \) and the coordinates of the point on the unit circle are \( \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \).

We can use these coordinates to find the values of the three trigonometric functions as follows:

1. **Sine (sin)**: In the unit circle, the sine of the angle \( t \) is the y-coordinate of the point where the terminal side of the angle intersects the unit circle.
   \[
   \sin t = \frac{1}{2}
   \]

2. **Tangent (tan)**: The tangent of the angle \( t \) is given by the ratio of the y-coordinate to the x-coordinate of the point.
   \[
   \tan t = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
   \]

3. **Secant (sec)**: The secant function is the reciprocal of the cosine function. The cosine of the angle \( t \) is the x-coordinate of the point.
   \[
   \sec t = \frac{1}{\cos t} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
   \]

Thus, the exact values are:

- \( \sin t = \frac{1}{2} \)
- \( \tan t = \frac{\sqrt{3}}{3} \)
- \( \sec t = \frac{2\sqrt{3}}{3} \)

Would you like more details on how these values were derived or have any specific questions?

Here are five related questions to further explore:
1. How is the unit circle used to define trigonometric functions?
2. What is the relationship between sine, cosine, and the coordinates on the unit circle?
3. Can you explain why \( \tan t = \frac{y}{x} \) on the unit circle?
4. How would you find the cotangent and cosecant of this angle?
5. What is the significance of the angle \( t \) in radians versus degrees?

**Tip**: Always remember that the unit circle has a radius of 1, which simplifies the definitions of sine and cosine to the coordinates \( (x, y) \) on the circle.

Use the angle in the unit circle to find the value of the three trigonometric functions.

This trigonometry problem demonstrates how to find the values of sine, tangent, and secant using the unit circle. With exact values calculated for an angle t, the problem covers core concepts of the unit circle, including sine, cosine, and secant definitions. Suitable for Grades 10-12.

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