Given that point $A$ on the unit circle has coordinates $\left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right)$, we can calculate the six trigonometric functions as follows:

1. Sine ($\sin$): $\sin(\theta) = y$ $\sin(\theta) = \frac{1}{2}$

2. Cosine ($\cos$): $\cos(\theta) = x$ $\cos(\theta) = -\frac{\sqrt{3}}{2}$

3. Tangent ($\tan$): $\tan(\theta) = \frac{y}{x}$ $\tan(\theta) = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

4. Cosecant ($\csc$): $\csc(\theta) = \frac{1}{y}$ $\csc(\theta) = \frac{1}{\frac{1}{2}} = 2$

5. Secant ($\sec$): $\sec(\theta) = \frac{1}{x}$ $\sec(\theta) = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}$

6. Cotangent ($\cot$): $\cot(\theta) = \frac{x}{y}$ $\cot(\theta) = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3}$

Summarizing the values of the trigonometric functions:

1. $\sin(\theta) = \frac{1}{2}$
2. $\cos(\theta) = -\frac{\sqrt{3}}{2}$
3. $\tan(\theta) = -\frac{\sqrt{3}}{3}$
4. $\csc(\theta) = 2$
5. $\sec(\theta) = -\frac{2\sqrt{3}}{3}$
6. $\cot(\theta) = -\sqrt{3}$

Would you like more details or have any questions?

Relative Questions:

1. What are the coordinates of point $A$ if it is at $\frac{\pi}{6}$ radians?
2. How do the trigonometric functions change if point $A$ is at $\frac{5\pi}{6}$ radians?
3. What are the values of the trigonometric functions for point $A$ at $\frac{7\pi}{6}$ radians?
4. Explain the behavior of $\sec(\theta)$ and $\csc(\theta)$ for point $A$ on the unit circle.
5. What are the six trigonometric function values for a point $B$ at $\frac{\pi}{4}$ radians?
6. How do you determine the angle $\theta$ corresponding to the given coordinates on the unit circle?
7. What is the significance of the signs of the trigonometric functions in different quadrants?
8. How can you use the unit circle to find the values of trigonometric functions for negative angles?

Tip:

When working with trigonometric functions, it's helpful to remember that each function's value is tied to a specific coordinate pair on the unit circle. Knowing the coordinates makes it straightforward to compute the function values.