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The concept of slope is fundamental in understanding the characteristics of a line in a coordinate plane. The slope is a measure of the steepness or angle of a line and is usually represented by the letter \(m\). It describes how much the y-value of a line increases or decreases as the x-value increases by one unit. In practical terms:

• A positive slope means the line ascends from left to right.
• A negative slope means the line descends from left to right.
• A zero slope indicates a horizontal line.
• An undefined slope refers to a vertical line.

For the line equation \(y = 3x - 4\), the slope is \(3\). This means for every one unit increase in \(x\), \(y\) increases by three units. Recognizing the slope's role is key to finding the line’s inclination, as it connects directly to its tangent.

The arctangent (denoted as \(\arctan\)) is an inverse trigonometric function used primarily to find the angle whose tangent is a given number. In this exercise, the tangent of the angle formed by the line with the x-axis, or its slope, is known to be \(3\). To find the angle \(\theta\) of inclination, we use the arctangent function:\[\theta = \arctan(3)\]The arctan function will return \(\theta\) in radians by default, producing the angle relative to the x-axis. This step is crucial as it translates the slope, a simple ratio, into an observable angle. This angle helps not only in understanding the line's steepness but also in visualizing how it "sits" on the graph.

Often angles are required in degrees rather than radians for easier interpretation and use in practical applications. Once we've found the angle \(\theta\) in radians from the \(\arctan(3)\), the next step is converting it into degrees. This conversion process involves a simple mathematical operation:To convert radians to degrees, multiply by the conversion factor:\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]For example, if \(\theta\) in radians is calculated, the degrees equivalent can be found by multiplying the radian measure by \(\frac{180}{\pi}\). This process changes an abstract radian measure into degrees, making it accessible for broader contexts, such as navigation or construction.