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### Part 1: Fill in the missing degree and radian measurements.

The table asks for both degree and radian measures and the coordinates of the robot arm's end for angles \( 0^\circ \), \( 45^\circ \), and \( 120^\circ \).

For a robotic arm with a length of 1 unit, the coordinates of the end of the arm for any given angle \(\theta\) can be determined using:
\[
x = \cos(\theta)
\]
\[
y = \sin(\theta)
\]
where \(\theta\) is the angle in radians.

### Let's fill in the table for each angle.

1. \( 0^\circ \):
    - Radians: \( 0 \) 
    - Coordinates: \( (\cos(0), \sin(0)) = (1, 0) \)

2. \( 45^\circ \):
    - Radians: \( \frac{\pi}{4} \)
    - Coordinates: \( (\cos(\frac{\pi}{4}), \sin(\frac{\pi}{4})) = \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \)

3. \( 120^\circ \):
    - Radians: \( \frac{2\pi}{3} \)
    - Coordinates: \( (\cos(\frac{2\pi}{3}), \sin(\frac{2\pi}{3})) = \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right) \)

Let me update the table with these values.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, here is the information I can provide manually:

1. **Angle: \(0^\circ\)**
   - Radians: \(0\)
   - Coordinates: \((1, 0)\)

2. **Angle: \(45^\circ\)**
   - Radians: \(\frac{\pi}{4}\)
   - Coordinates: \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)
   - Approximation: \((0.707, 0.707)\)

3. **Angle: \(120^\circ\)**
   - Radians: \(\frac{2\pi}{3}\)
   - Coordinates: \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)
   - Approximation: \((-0.5, 0.866)\)

If you want more detailed explanations, feel free to ask.

Would you like help sketching the robot arm positions or solving anything else?

### Relative Questions:
1. How do trigonometric functions relate to the movement of robotic arms?
2. How can the Pythagorean theorem help when calculating the robot's arm position?
3. What are the real-world applications of calculating the angles and positions of robotic arms?
4. How does understanding radians benefit engineers in the field of robotics?
5. How would the coordinates change if the arm's length was increased?

### Tip:
When dealing with angles in real-world engineering problems, always ensure that you're comfortable switching between degrees and radians, as different systems use them interchangeably!

Determine the position of the end of a robotic arm of length one unit for various angles, given in both degrees and radians, using trigonometric functions.

Learn how to find the position of a robotic arm of length one unit using trigonometric functions at different angles. This problem covers the use of radians and cosine/sine functions to calculate coordinates in the plane. Suitable for students in Grades 10-12.

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