91Ó°ÊÓ

The slope of a line is a fundamental concept in geometry and helps to describe the steepness or direction of a line. It is denoted by the letter "m" and is calculated as the ratio of vertical change (rise) to horizontal change (run) between two points on the line.
For example, if you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope "m" can be found using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

When the slope is positive, it indicates an upward direction from left to right; when it's negative, the line moves downward. A slope of 1, as in the exercise, tells us that for each unit of horizontal movement, the vertical movement is also one unit.
Lines with a slope of 1 form a special angle of 45 degrees or \( \pi/4 \) radians with the horizontal axis. This provides a visual and intuitive sense of symmetry and balance, as both axes contribute equally.

Converting angles from radians to degrees is essential as we often encounter angles expressed in both units. The basic relation between radians and degrees is crucial for this conversion. One complete circle in radians is \( 2\pi \) and in degrees is 360, establishing the equivalence:

• \( 2\pi \) radians = 360 degrees
• Thus, \( 1 \) radian = \( \frac{180}{\pi} \) degrees

To convert an angle from radians to degrees, you multiply the radian value by \( \frac{180}{\pi} \).
In our exercise, the angle \( \pi/4 \) radians was converted to degrees as follows:

• \( \frac{\pi}{4} \times \frac{180}{\pi} = 45 \) degrees

This formula ensures a straightforward and accurate conversion. Understanding this conversion helps bridge the gap between different mathematical contexts where either radians or degrees might be used predominantly.

The arctangent function, often written as \( \tan^{-1}(x) \) or \( \arctan(x) \), is the inverse of the tangent function. It is used to find the angle whose tangent is a given number.
The function returns the angle \( \theta \) in radians such that:

• \( \tan(\theta) = x \)

Using a calculator or mathematical software, one can easily find this angle. In our exercise, the slope was 1, meaning:

• \( \theta = \tan^{-1}(1) \)
• The result is \( \pi/4 \), or 0.7854 radians

This result is intuitive because \( \tan(\pi/4) = 1 \), revealing a perfect match between the function's behavior and the geometric interpretation of angles.