Problem 29 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line passing through the points. $$(6,1),(10,8)$$

Step-by-Step Solution

Answer

The inclination of the line passing through the points (6,1) and (10,8) is approximately 1.0516 radians or 60.2551 degrees.

Step 1: Calculate the Slope of the Line

The first step is to calculate the slope of the line. The slope formula is given as \(m = \frac{y2 - y1}{x2 - x1}\). Here, (6,1) can be (x1, y1) and (10,8) can be (x2, y2). Plugging these values into the formula gives \(m = \frac{8 - 1}{10 - 6} = 1.75\)

Step 2: Calculate Angle of Elevation in Radians

The angle \(\theta\) the line makes with the x-axis (angle of elevation) in radians can be calculated using the arctangent function ('atan'). Here, \(m\) is the slope from step 1. The formula is given as \(\theta = atan(m)\). Substituting \(m = 1.75\) gives \(\theta = atan(1.75)\). You can calculate this using a scientific calculator. For our example, \(\theta\) results in approximately 1.0516 radians.

Step 3: Calculate Angle of Elevation in Degrees

To express the angle of elevation in degrees, the radians must be converted using the formula \(\text{Degree} = \theta \times \frac{180}{\pi}\). Substituting \(\theta = 1.0516\) radians into the formula results in roughly 60.2551 degrees

Key Concepts

Slope of a Line

Understanding the concept of the slope of a line is fundamental in geometry and algebra. The slope is essentially a measure of how steep a line is. It is calculated by determining the vertical change (rise) between two points on the line, divided by the horizontal change (run). Mathematically, this is expressed with the familiar formula, which takes two points \( (x1, y1) \) and \( (x2, y2) \)—the coordinates on a Cartesian plane:
\[ m = \frac{y2 - y1}{x2 - x1} \]
To illustrate, let's take two points on a line: \( (6, 1) \) and \( (10, 8) \). Plugging these values into the formula gives the slope \( m \) as follows:
\[ m = \frac{8 - 1}{10 - 6} = 1.75 \]
The slope indicates that for every unit you move to the right along the x-axis, the line rises 1.75 units. This ratio translates well when you're trying to sketch graphs or analyze linear trends in data. Remember, a positive slope indicates an upward trend, while a negative slope indicates a downward trend.

Radians to Degrees Conversion

When working with angles, it’s crucial to be able to convert between radians and degrees. Radians and degrees are two different units for measuring angles, just like meters and feet are for measuring length. In the context of the exercise, once you have calculated the angle of elevation in radians, you might need to express it in degrees, since degrees are commonly used in daily life.
The formula to convert radians to degrees is straightforward:
\[ \text{Degrees} = \theta \times \frac{180}{\pi} \]
Here, \( \pi \) (pi) approximates to 3.14159. So, an angle of \( 1.0516 \) radians can be converted to degrees as follows:
\[ \text{Degrees} = 1.0516 \times \frac{180}{\pi} \approx 60.2551 \]
This conversion is crucial in fields like trigonometry, physics, and engineering where precision in angle measurement is necessary. Converting radians to degrees helps bridge the gap between theoretical mathematics and its practical applications.

Arctangent Function

The arctangent function, often abbreviated as \( \text{atan} \), is one of the inverse trigonometric functions, specifically the inverse of the tangent function. In the context of slopes and angles involving right-angled triangles, the arctangent is used to find an angle when the opposite side and adjacent side lengths are known. The function outputs an angle in radians.
In our exercise, once the slope \( m \) of the line is calculated, the arctangent function gives us the angle \( \theta \) that the line makes with the x-axis:
\[ \theta = \text{atan}(m) \]
Substituting the slope value \( m = 1.75 \) into the arctangent function will yield the angle in radians. This is done using a scientific calculator or a computer algebra system:
\[ \theta = \text{atan}(1.75) \approx 1.0516 \text{ radians} \]
The arctangent function is especially useful because it considers the sign of both the numerator and the denominator of the slope ratio, thereby appropriately assigning the angle to the correct quadrant on a coordinate plane. Understanding the arctangent function is indispensable for students studying trigonometry, calculus, and for anyone who needs to solve problems involving angles and slopes.