Problem 18 - Precalculus with Limits

Question

In Exercises 13-18, find the inclination \(\theta\) (in radians and degrees) of the line with a slope of \(m\). \(m = -\frac{5}{2}\)

Step-by-Step Solution

Answer

Firstly, find the angle in radians using the slope with the arctan function on a calculator. Then, transform this angle from radians to degrees through the conversion factor \(\frac{180}{\pi}\).

Step 1: Input the slope into the arctangent function formula

Use the formula \(\theta = \arctan(m)\) to find the inclination of the line. Here, \(\arctan\) stands for the inverse of the tangent function, and \(m\) is the slope of the line. Substituting our given slope \(m = -\frac{5}{2}\), it becomes \(\theta = \arctan(-\frac{5}{2})\).

Step 2: Compute the arctangent of the slope

By inputting this into a scientific calculator, it provides \(\theta\) in radians. Be sure to use the mathematical convention of considering counter-clockwise angles as positive and clockwise angles as negative.

Step 3: Convert radians to degrees

After finding the angle in radians, convert it to degrees using the conversion factor \(\frac{180}{\pi}\) degrees is equal to 1 radian. Multiply the radian measurement by the conversion factor to get the angle in degrees. Be sure to take care that negative angles represent a clockwise direction.

Key Concepts

Arctangent Function

The arctangent function, commonly denoted as \( \arctan \theta \), is the inverse function of the tangent. In simpler terms, if you know the slope \( m \) of a line, the arctangent helps you find the angle \( \theta \) that the line makes with the horizontal axis. This angle is often referred to as the line's inclination. Working with arctangent is straightforward. You simply input the slope into your calculator or use a mathematical table to find the angle in radians.
The relationship is given by the formula:

• \( \theta = \arctan(m) \)

For example, if the slope \( m \) is \(-\frac{5}{2}\), then \( \theta = \arctan(-\frac{5}{2}) \). It is important to understand that the result from \( \arctan \) yields angles in radians. So, whenever you calculate an arctangent, you're getting an angle compatible with circle measurements expressed through radians. Since radians are the standard unit for measuring angles in mathematics, they'll be quite useful across different math topics.

Slope of a Line

A line's slope \( m \) fundamentally describes how steep the line is. Slope is determined by the rise over the run, which means how much the line goes up or down for every horizontal step taken. It's given as:

• \( m = \frac{\text{Rise}}{\text{Run}} \)

If the slope is positive, the line rises as you move right. If it is negative, the line falls as you move right. Understanding slope is essential because it provides a straightforward way to describe the direction and steepness of a line on a coordinate plane.

When calculating the inclination of a line, the slope tells us exactly how to orient the line on the plane. For instance, a slope of \(-\frac{5}{2}\) means the line falls steeply downwards as you move from left to right. It's worth noting that horizontal lines have a slope of zero, and vertical lines have an undefined slope due to division by zero in the slope formula. This characteristic of slope helps us predict how lines behave when graphed.

Radians to Degrees Conversion

In many mathematical contexts, radians are used as the default unit for angles, predominantly because they provide a direct way to link angles to arc lengths on a circle. However, degrees are more intuitive for most people.
Converting radians to degrees involves a simple multiplication by a conversion factor. This is essential when you need an angle expressed in degrees for practical understanding. The conversion factor itself is derived from the equivalence between a full circle measured in radians and degrees, where:

• \( 180^{\circ} = \pi \text{ radians} \)

To convert, multiply the angle in radians by \( \frac{180}{\pi} \). For example, if you have an angle of \( \theta \) radians, the formula to convert is:

• \( \theta_{\text{degrees}} = \theta_{\text{radians}} \cdot \frac{180}{\pi} \)

This conversion is crucial, especially in fields like engineering, navigation, and computer graphics, where degrees are often used. Recognizing this interplay between radians and degrees ensures you can communicate and analyze angular measures effectively in different scenarios.