91Ó°ÊÓ

Understanding the concept of the inverse tangent is crucial in precalculus, especially when dealing with line inclinations. The inverse tangent function, also known as arctangent or \( \tan^{-1} \), allows us to find the angle whose tangent is a given number. Tangent, in trigonometry, relates an angle in a right-angled triangle to the ratio of the length of the opposite side to the adjacent side. So, when we have the slope \(m\) of a line, which can be seen as this ratio, the inverse tangent gives us the angle \(\theta\) at which the line inclines, or 'tilts', from the horizontal axis.

Particularly, if a line has a slope of -2, as in the given exercise, taking the inverse tangent of this slope, \(\tan^{-1}(-2)\), reveals the angle of inclination in radians. It's important to note that calculators or software should be set to the correct mode—radian or degree—depending on what is desired. However, if not specified, the inverse tangent usually provides the angle in radians.

The slope of a line is a measure of how steep the line is and is a fundamental concept in algebra and precalculus. It's represented by the letter \(m\) and is calculated as the change in the y-coordinate divided by the change in the x-coordinate between two distinct points on the line (also referred to as 'rise over run').

For a line defined by the equation \(y = mx + b\), \(m\) is the slope, and \(b\) is the y-intercept. If a slope is positive, the line rises as it moves to the right; if the slope is negative, the line falls. The greater the magnitude of the slope, the steeper the line. Reflecting on our example, a slope of -2 means that for every one unit we move to the right along the horizontal axis, we move two units down. This slope not only dictates the direction but also the angle of the line with respect to the horizontal, which can be determined using the inverse tangent as described earlier.

After determining the inclination of a line in radians using the inverse tangent, often there is a need to interpret this angle in degrees. To convert radians to degrees, we use the fact that \(180^\circ\) is equivalent to \(\pi\) radians. The conversion formula is given by multiplying the radians by \(\frac{180}{\pi}\).

When carrying out radian to degree conversion, it's handy to remember that \(\pi \approx 3.14159\), so for the exercise at hand, after finding the inclination in radians using \(\tan^{-1}(-2)\), we multiply this result by \(\frac{180}{\pi}\) to express our angle in degrees. This step is crucial for those who are more familiar with the degree system or when the context requires an answer in degrees, for example in navigation or engineering.