Problem 34 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line passing through the points. $$\left(\frac{2}{5},-\frac{3}{4}\right),\left(-\frac{11}{10},-\frac{1}{4}\right)$$

Step-by-Step Solution

Answer

The inclination of the line passing through the given points in radians is \(\arctan(-10/27)\) and in degrees, it is \(\left(\arctan(-10/27)\right) \times \frac{180}{\pi}\) degrees.

Step 1: Identify the coordinates of the two points

The points given are \((2/5, -3/4)\) and \((-11/10, -1/4)\). So, we identify \(x_1 = 2/5\), \(y_1 = -3/4\), \(x_2 = -11/10\) and \(y_2 = -1/4\).

Step 2: Compute the slope using the formula

We use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given values, we get \(m = \frac{-1/4 - (-3/4)}{-11/10 - 2/5} = \frac{1/2}{-27/10} = -10/27\).

Step 3: Calculate the inclination in radians

The inclination \(\theta\) is the arctangent of the slope. So, \(\theta = \arctan(m) = \arctan(-10/27)\).

Step 4: Convert the inclination to degrees

Converting radians to degrees, we get \(\theta = \left(\arctan(-10/27)\right) \times \frac{180}{\pi}\) degrees.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in geometry and algebra, describing how steep the line is. It indicates the rate at which the line rises or falls.
For any two points on a line,

• The slope is determined by dividing the difference in the y-values by the difference in the x-values.
• This is mathematically expressed as the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points on the line.
• A positive slope indicates the line is rising, while a negative slope shows it is falling.
• If the slope is zero, the line is horizontal, and an undefined slope suggests a vertical line.

Knowing the slope is crucial to finding the inclination of the line, which connects to the angle it makes with the horizontal axis.

Arctangent Function

The arctangent function, denoted as \( \arctan \), is an inverse trigonometric function.
It is used to find the angle whose tangent is a given number. This calculation is key when we're looking for the angle of inclination of a line with a given slope.

• Given a slope \( m \), the angle \( \theta \) of inclination is \( \theta = \arctan(m) \).

The arctangent function returns an angle in radians.

• This angle can range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), covering all possible inclinations of a line.
• Essentially, \( \arctan \) helps you find the angle between the line and the x-axis, providing insight into how the line is oriented in the coordinate plane.

Understanding how \( \arctan \) works is important for converting that angle into other units, such as degrees.

Conversion between Radians and Degrees

Angles can be expressed in either radians or degrees.
Converting between these two is a common task in mathematics, especially when dealing with trigonometric functions and geometric calculations.

• To convert an angle from radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \).

For example, if you find \( \theta \) in radians using the arctangent function as in \( \theta = \arctan(-10/27) \), you would then convert it to degrees by:

• Calculating \( \theta \times \frac{180}{\pi} \).

This conversion is crucial since most day-to-day angle measures are given in degrees.
Understanding this conversion helps bridge the gap between abstract mathematical concepts and practical applications, making it easier to relate mathematical insights to real-world situations.