Sketch an angle theta in standard position such that theta has the least possible positive​ measure, and the point ​(negative 4​,0​) is on the terminal side of theta. Then find the values of the six trigonometric functions for the angle. Rationalize denominators if applicable. Do not use a calculator.
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Part 1
Choose the correct graph below.


To approach this problem, let's first interpret it step by step.

1. **Sketching the angle \(\theta\):**
   - The point \((-4, 0)\) lies on the negative side of the x-axis.
   - An angle in **standard position** starts with its initial side on the positive x-axis. The **terminal side** of the angle passes through the point \((-4, 0)\).
   - To get to \((-4, 0)\) from the positive x-axis, we must rotate **180 degrees** counterclockwise. Thus, the least possible positive measure of \(\theta\) is \(180^\circ\) or \(\pi\) radians.

2. **Six trigonometric functions of the angle \(\theta = 180^\circ\):**
   Since the point \((-4, 0)\) is on the terminal side, we can now find the values of the six trigonometric functions. We know that:
   - \(r = \sqrt{(-4)^2 + 0^2} = \sqrt{16} = 4\) (the distance from the origin to the point is the radius \(r\)).
   
   Now, the trigonometric values are as follows:
   
   - \(\sin(\theta) = \frac{y}{r} = \frac{0}{4} = 0\)
   - \(\cos(\theta) = \frac{x}{r} = \frac{-4}{4} = -1\)
   - \(\tan(\theta) = \frac{y}{x} = \frac{0}{-4} = 0\)
   
   The reciprocal functions are:
   
   - \(\csc(\theta) = \frac{1}{\sin(\theta)} = \text{undefined} \) (since \(\sin(\theta) = 0\))
   - \(\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{-1} = -1\)
   - \(\cot(\theta) = \frac{1}{\tan(\theta)} = \text{undefined} \) (since \(\tan(\theta) = 0\))

### Correct graph:
- The correct graph would show the angle starting on the positive x-axis, with the terminal side extending through the point \((-4, 0)\), which is exactly 180 degrees.

Would you like me to generate this graph for you?

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**Relative questions:**
1. What is the significance of standard position when sketching an angle?
2. How do you calculate the distance \(r\) from the origin to a point in trigonometry?
3. Why are some trigonometric functions undefined for certain angles?
4. How do reciprocal trigonometric functions relate to their basic functions?
5. What are the key characteristics of angles that lie on the x-axis?

**Tip:** Always check if the point given lies on one of the axes, as this simplifies trigonometric function calculations significantly.

Sketch an angle theta in standard position such that theta has the least possible positive measure, and the point (negative 4, 0) is on the terminal side of theta. Then find the values of the six trigonometric functions for the angle. Rationalize denominators if applicable. Do not use a calculator.

Learn how to sketch an angle in standard position with the point (-4, 0) on the terminal side and calculate the six trigonometric functions for the angle. This detailed guide covers key concepts such as trigonometric functions, radian measure, and reciprocal functions, suitable for high school students.

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