Given a slope y=mx, where m=tan(theta) and theta=tan^-1(m), how can I find a second slope, we'll call it m2, such that the angle of m2 is equal to half that of m? Let me repeat, I am not asking for half the slope, but half the angle of the slope.

To find a second slope \( m_2 \), such that the angle of \( m_2 \) is half that of \( m \), follow these steps:

### 1. Understanding the problem:
Given:
- The slope of a line is \( m \), where \( m = \tan(\theta) \).
- You are asked to find \( m_2 \), the slope of a new line, where the angle \( \theta_2 \) of \( m_2 \) is half of \( \theta \), i.e., \( \theta_2 = \frac{\theta}{2} \).

### 2. Use the relationship between slope and angle:
- The slope of a line is related to its angle by \( m = \tan(\theta) \).
- For the new line, we know the angle is \( \theta_2 = \frac{\theta}{2} \), so its slope is:
  \[
  m_2 = \tan\left(\frac{\theta}{2}\right)
  \]

### 3. Express \( m_2 \) in terms of \( m \):
Now, we need to express \( \tan\left(\frac{\theta}{2}\right) \) in terms of \( m \), which is \( \tan(\theta) \). To do this, we use the **half-angle identity** for the tangent function:
\[
\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)}
\]
Alternatively, using the expression involving \( m \), the half-angle formula for tangent can be written as:
\[
\tan\left(\frac{\theta}{2}\right) = \frac{m}{1 + \sqrt{1 + m^2}}
\]
Thus, the slope of the new line \( m_2 \) is:
\[
m_2 = \frac{m}{1 + \sqrt{1 + m^2}}
\]

### Conclusion:
The slope \( m_2 \) of the new line, where the angle is half of the original line's angle, is given by:
\[
m_2 = \frac{m}{1 + \sqrt{1 + m^2}}
\]

Would you like more details on the derivation or any clarification?

### Relative Questions:
1. How do we derive the half-angle identity for tangent from basic trigonometric principles?
2. What is the geometrical interpretation of halving the angle of a line's slope?
3. Can we generalize this concept to other trigonometric functions, like sine or cosine?
4. What happens to \( m_2 \) as \( m \) approaches very large values or zero?
5. How does this formula behave in the case of negative slopes?

### Tip:
For small angles, the slope of a line approximates the angle itself (in radians), so for very small values of \( \theta \), halving the angle roughly halves the slope too!

Discover how to calculate the slope of a line where the angle is half that of a given slope. This problem explores trigonometric principles, including the half-angle identity for tangent, and provides a formula for solving in terms of m. Suitable for Grades 10-12 students.

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