I have a 1-meter long segment, placed on a plane at a random angle. What is the expected length of its projection on the X axis?

To solve the problem of finding the expected length of the projection of a 1-meter segment onto the X-axis, we can approach it using the concept of the **expected value** in probability and trigonometry.

### Problem Breakdown

1. **Segment on the Plane**: The segment is 1 meter long and is placed at a random angle \(\theta\) on a plane.

2. **Projection on the X-Axis**: The length of the projection of the segment on the X-axis is given by \(\text{Projection Length} = \cos(\theta)\), where \(\theta\) is the angle between the segment and the X-axis.

3. **Expected Value**: We want to find the expected value of \(\cos(\theta)\) over all possible angles \(\theta\), which range from 0 to \(2\pi\) (since the angle is random and uniformly distributed).

### Calculation

The expected value \(E[\cos(\theta)]\) of the cosine function over all angles from 0 to \(2\pi\) can be calculated as:

\[
E[\cos(\theta)] = \frac{1}{2\pi} \int_0^{2\pi} \cos(\theta) \, d\theta
\]

Let's calculate this integral.

### Integral Solution

\[
\int_0^{2\pi} \cos(\theta) \, d\theta = \left[\sin(\theta)\right]_0^{2\pi} = \sin(2\pi) - \sin(0) = 0 - 0 = 0
\]

So, the expected value is:

\[
E[\cos(\theta)] = \frac{1}{2\pi} \times 0 = 0
\]

### Conclusion

The expected length of the projection of a 1-meter segment onto the X-axis is **0 meters**.

This result makes intuitive sense because over a full rotation, the positive and negative projections cancel out, leading to an expected value of zero.

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Would you like more details or have any questions on this? Here are some related questions:

1. What is the expected value of the absolute value of the projection?
2. How does the expected projection change if the segment length varies?
3. Can you explain the significance of symmetry in calculating expected values?
4. What is the expected value of the projection on the Y-axis?
5. How would the expected value change if the segment was not uniformly distributed in angle?
6. What happens to the expected value if the segment is constrained to a specific angular range?
7. Can this problem be extended to 3D projections on a plane?
8. How does this concept apply to real-world problems like signal analysis?

**Tip**: Understanding symmetry can simplify the calculation of expected values in many problems involving random distributions.

Understand the expected length of the projection on the X-axis for a 1-meter segment placed at a random angle on a plane. Learn how to calculate the expected value using probability and trigonometry.

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