After completing one revolution starting from the initial line, point A is observed to have coordinates () on the unit circle. Using that information, please answer the following questions:

Calculate all 6 values of the trigonometric functions (clearly state the formulae used to calculate these functions).

To calculate the six trigonometric functions based on the coordinates of point \( A \) on the unit circle after completing one revolution, we first need to establish the coordinates of point \( A \). Let's denote the coordinates of point \( A \) as \((x, y)\).

Since point \( A \) lies on the unit circle, the coordinates \((x, y)\) must satisfy the equation of the unit circle:
\[ x^2 + y^2 = 1 \]

The six trigonometric functions are defined as follows based on the coordinates \((x, y)\):

1. **Sine (\(\sin\))**:
   \[ \sin(\theta) = y \]

2. **Cosine (\(\cos\))**:
   \[ \cos(\theta) = x \]

3. **Tangent (\(\tan\))**:
   \[ \tan(\theta) = \frac{y}{x} \]
   Note: \(\tan(\theta)\) is undefined when \( x = 0 \).

4. **Cosecant (\(\csc\))**:
   \[ \csc(\theta) = \frac{1}{y} \]
   Note: \(\csc(\theta)\) is undefined when \( y = 0 \).

5. **Secant (\(\sec\))**:
   \[ \sec(\theta) = \frac{1}{x} \]
   Note: \(\sec(\theta)\) is undefined when \( x = 0 \).

6. **Cotangent (\(\cot\))**:
   \[ \cot(\theta) = \frac{x}{y} \]
   Note: \(\cot(\theta)\) is undefined when \( y = 0 \).

Let's assume point \( A \) has coordinates \( (1, 0) \), which is a common result after completing one revolution starting from the initial line (the positive \( x \)-axis).

Using these coordinates \((x, y) = (1, 0)\):

1. **Sine (\(\sin(\theta)\))**:
   \[ \sin(\theta) = y = 0 \]

2. **Cosine (\(\cos(\theta)\))**:
   \[ \cos(\theta) = x = 1 \]

3. **Tangent (\(\tan(\theta)\))**:
   \[ \tan(\theta) = \frac{y}{x} = \frac{0}{1} = 0 \]

4. **Cosecant (\(\csc(\theta)\))**:
   \[ \csc(\theta) = \frac{1}{y} \]
   Since \( y = 0 \), \(\csc(\theta)\) is undefined.

5. **Secant (\(\sec(\theta)\))**:
   \[ \sec(\theta) = \frac{1}{x} = \frac{1}{1} = 1 \]

6. **Cotangent (\(\cot(\theta)\))**:
   \[ \cot(\theta) = \frac{x}{y} \]
   Since \( y = 0 \), \(\cot(\theta)\) is undefined.

Therefore, the values of the trigonometric functions are:

1. \(\sin(\theta) = 0\)
2. \(\cos(\theta) = 1\)
3. \(\tan(\theta) = 0\)
4. \(\csc(\theta) = \text{undefined}\)
5. \(\sec(\theta) = 1\)
6. \(\cot(\theta) = \text{undefined}\)

Would you like more details or have any questions?

### Relative Questions:
1. What are the coordinates of point \( A \) if it is at \(\pi/2\) radians?
2. How do the trigonometric functions change if point \( A \) is at \(\pi\) radians?
3. What are the values of the trigonometric functions for point \( A \) at \(3\pi/2\) radians?
4. Explain the behavior of \(\tan(\theta)\) and \(\cot(\theta)\) at \(\pi/2\) and \(3\pi/2\).
5. What are the six trigonometric function values for a point \( B \) at \(\pi/4\) radians?
6. How do you find the trigonometric functions for any point \( (x, y) \) on the unit circle?
7. What is the significance of the unit circle in trigonometry?
8. How do the signs of trigonometric functions change in different quadrants?

### Tip:
Remember that the unit circle is a powerful tool for visualizing and understanding the behavior of trigonometric functions across different angles.

Learn how to calculate the six trigonometric functions (sin, cos, tan, csc, sec, cot) based on the coordinates of point A on the unit circle after completing one revolution. Detailed explanation with formulas and step-by-step solution.

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