Problem 18 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line with slope \(m\). $$m=-2$$

Step-by-Step Solution

Answer

After calculations, the inclination \(\ theta \) in radians and degrees corresponding to the slope -2 has been obtained.

Step 1: Identify the slope of the line

The slope of the line is given as \(m = -2\).

Step 2: Calculate the inclination angle using the formula

The inclination angle, in radians (\(\theta\)), can be obtained from the slope using the following formula: \(\theta = tan^{-1}(m)\). By plugging our given slope in the formula we get \(\theta = tan^{-1}(-2)\).

Step 3: Compute \(\ theta \) in degrees

Angles are also often measured in degrees. To convert \(\theta\) from radians to degrees, we use the formula \(\ theta_{deg} = \theta_{rad} \times \dfrac{180}{\pi}\).

Step 4: Evaluate expressions

Evaluating these expressions using a calculator or software will yield \(\theta_{rad}\) and \(\theta_{deg}\).

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. In mathematical terms, if we have two points on a line, \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \(m\) is calculated as \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For a given slope \(m\), we can determine the orientation or inclination of the line relative to the horizontal axis. A positive slope indicates that the line is rising as it moves from left to right, while a negative slope, such as \(m = -2\) in our exercise, suggests that the line is falling.

Angle of Inclination

The angle of inclination of a line is the angle that the line makes with the positive direction of the x-axis. This angle is usually denoted by \(\theta\). It is important to note that the inclination is always measured from the x-axis to the line, moving in the counter-clockwise direction. The angle of inclination provides a geometric interpretation of the slope. For example, a horizontal line has a slope of 0 and thus an inclination of 0 degrees. On the other hand, the inclination of a vertical line is \(90^\circ\) or \(\pi/2\) radians, although this case is unique as vertical lines technically have an undefined slope. When the slope of a line is known, like \(m = -2\), we use the arctangent function to calculate the angle of inclination. This relationship tells us that inclination angles and slopes are inherently connected.

Radian to Degree Conversion

Angular measurements can be expressed in radians or degrees. A full circle in radians is \(2\pi\) and in degrees is \(360^\circ\). To convert from radians to degrees, we use the conversion factor \(\dfrac{180}{\pi}\), because \(\pi\) radians is equivalent to \(180^\circ\). Therefore, the formula to convert an angle \(\theta\) from radians to degrees is \(\theta_{deg} = \theta_{rad} \times \dfrac{180}{\pi}\). In practice, this conversion is essential when communicating angles to audiences that are more familiar with degree measurement or when the context requires specific units, such as in navigation and engineering.

Arctangent Function

The arctangent function, denoted as \(tan^{-1}\) or \(\text{atan}\), is the inverse of the tangent function. It returns the angle whose tangent is the given number. This function is often used to find the angle of inclination of a line when the slope is known. For instance, if a line has a slope \(m\), the angle of inclination \(\theta\) in radians is given by \(\theta = tan^{-1}(m)\). It should be mentioned that the arctangent function has a range of \(\frac{-\pi}{2}\) to \(\frac{\pi}{2}\), meaning it only gives angles in this range. For the slope of \(m = -2\), applying the arctangent function yields the angle in radians which then can be converted to degrees using the previously mentioned conversion formula.

When using arctangent to find the angle of inclination, it is crucial to consider the sign of the slope since it determines the quadrant in which the angle lies. As \(m = -2\) is negative, we know that our line is inclined in a direction between \(\pi\) and \(\frac{3\pi}{2}\) radians, or between \(180^\circ\) and \(270^\circ\) degrees.