Problem 22 - Precalculus with Limits

Question

In Exercises 19-26, find the inclination \(\theta\) (in radians and degrees) of the line passing through the points. \((3, \sqrt{3})\), \((6, -2\sqrt{3})\)

Step-by-Step Solution

Answer

The inclination \(\theta\) (in radians and degrees) of the line passing through the points \((3, \sqrt{3})\) and \((6, -2\sqrt{3})\) can be found by first calculating the slope, then finding the inclination in radians, and finally converting the radian measure into degrees. The specific values will depend on the calculations made in the process.

Step 1: Calculation of Slope

Calculate the slope \(m\) using the coordinates of the two points. Use the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). Substitute \(x_1 = 3\), \(y_1 = \sqrt{3}\), \(x_2 = 6\), and \(y_2 = -2\sqrt{3}\) into the formula and simplify to find the slope.

Step 2: Calculation of Inclination in Radians

Calculate the inclination angle \(\theta\) (in radians) using the slope. The formula to calculate the inclination \(\theta\) (in radians) is given by: \(\theta = atan(m)\). Substitute the calculated slope into the formula and simplify to find the inclination in radians.

Step 3: Conversion of Inclination into Degrees

Convert the inclination from radians to degrees. Use the conversion formula: degrees = radians * \(180 / \pi\). Replace 'radians' with the previously calculated inclination in radians and simplify to find the inclination in degrees.

Key Concepts

Slope of a Line

The slope of a line is a numerical measure of the line's steepness or incline. It is often denoted by the letter 'm' and mathematically described as the change in the y-coordinate (vertical change) divided by the change in the x-coordinate (horizontal change) between two distinct points on the line.

To compute the slope, we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For instance, if we have two points on the line \( P_1(3, \sqrt{3}) \) and \( P_2(6, -2\sqrt{3}) \), we can calculate the slope by subtracting the y-coordinate of \( P_1 \) from that of \( P_2 \) and dividing the result by the difference in their x-coordinates. It's critical to correctly apply this formula, as the slope is the foundation for understanding a line's behavior and subsequently determining the inclination angle.

Inclination Angle in Radians

Once the slope 'm' of the line is established, we can move forward to find the line's inclination angle in radians. An inclination angle is the angle a line makes with the positive direction of the x-axis.

The angle \( \theta \) in radians can be found using the arctangent function, denoted as \(\text{atan}\). The arctangent function is the inverse of the tangent function and is used because the slope of a line is the tangent of the inclination angle. The formula for calculating the inclination \( \theta \) in radians based on the slope is: \[ \theta = \text{atan}(m) \] This equation is applied to the slope calculated from two given points on the line to determine the line's angle of inclination relative to the x-axis.

Conversion of Radians to Degrees

Understanding angles in different measurement units is important, and we often need to switch between radians and degrees. Since the radian is the standard unit of angular measure in mathematics due to its natural emergence from the properties of circles, conversion is common.

To convert an angle from radians to degrees, we implement the following formula: \[ \text{degrees} = \text{radians} \times \left(\frac{180}{\pi}\right) \] If an inclination \( \theta \) given is in radians, this formula is used to express it in degrees, where \( \pi \) is approximately 3.14159. This conversion is essential when you want to communicate an angle in a more universally understood unit, such as when drawing angles on a graph or in various practical applications where the degree is the preferred unit of measure.