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.