Problem 23 - Precalculus with Limits

Question

In Exercises 19-26, 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 given by \( atan(\frac{7}{4}) \) radians or \( atan(\frac{7}{4}) \cdot \frac{180}{\pi} \) degrees.

Step 1: Calculate the slope of the line

Using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), with given points as \((6, 1)\) and \((10, 8)\), calculate \(m = \frac{8 - 1}{10 - 6} = \frac{7}{4}\). This gives the slope of the line.

Step 2: Calculate the inclination in radians

Using the formula \( \theta = atan(m) \), where \(m\) is the slope of the line, calculate \( \theta = atan(\frac{7}{4})\). This gives the inclination of the line in radians.

Step 3: Convert the inclination into degrees

Multiply \( \theta \) by \( \frac{180}{\pi} \) to convert the inclination from radians into degrees.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness. It tells us how much the line rises or falls as it moves from left to right. Mathematically, it's represented by the letter \( m \) and is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula finds the difference in the \( y \)-coordinates (vertical change) divided by the difference in the \( x \)-coordinates (horizontal change) between two points on the line.

• For the points \((6, 1)\) and \((10, 8)\), the slope \( m \) is:
  \( m = \frac{8 - 1}{10 - 6} = \frac{7}{4} \).
• This means that for every 4 units the line moves horizontally, it rises 7 units vertically.

Understanding the slope is crucial as it determines the angle or incline of the line.

Angle Conversion

In geometry, angles can be expressed in both degrees and radians. Conversion between these units is essential for various calculations.

• A full circle is \( 360 \) degrees, which is equivalent to \( 2\pi \) radians.
• To convert an angle from radians to degrees, use the formula:
  \[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]

For instance, when we find an inclination angle in radians, like \( \theta = \text{atan}\left(\frac{7}{4}\right) \), converting to degrees aids in understanding how steep the line appears in more familiar terms.
This unit conversion ensures consistency in mathematical and real-world applications.

Arctangent Function

The arctangent function, denoted as \( \text{atan} \) or \( \tan^{-1} \), is the inverse of the tangent function. It helps in finding the angle whose tangent is a given number.

• It is especially useful when calculating the inclination of a line based on its slope.
• The formula is \( \theta = \text{atan}(m) \), where \( m \) is the slope.

When you calculate \( \theta = \text{atan}\left(\frac{7}{4}\right) \), you are determining the angle that has a tangent value of \( \frac{7}{4} \).
This angle, \( \theta \), is the inclination, and knowing it helps in understanding the directional bias of the line relative to the horizontal axis.