Problem 17 - Precalculus with Limits

Question

In Exercises 13-18, find the inclination \(\theta\) (in radians and degrees) of the line with a slope of \(m\). \(m = \frac{3}{4}\)

Step-by-Step Solution

Answer

The incline of the line, in radians, is \(\theta_{radians} = \arctan(\frac{3}{4})\) and in degrees is \(\theta_{degrees} = \theta_{radians} × \frac{180}{\pi}\)

Step 1: Compute the angle in radians

First, calculate the angle of inclination in radians by using the inverse tangent function on the value of the slope \(m = \frac{3}{4}\). So, \(\theta_{radians} = \arctan(m) = \arctan(\frac{3}{4})\)

Step 2: Convert radians into degrees

To convert the angle from radians to degrees, use the formula \(degrees = radians × \frac{180}{\pi}\). Therefore, \(\theta_{degrees} = \theta_{radians} × \frac{180}{\pi}\)

Step 3: Plug into Calculator

Now, calculate the values by plugging the equations into the calculator to get the numerical values. After plugging in, you will get: \(\theta_{radians} = \arctan(\frac{3}{4})\) and \(\theta_{degrees} = \theta_{radians} × \frac{180}{\pi}\)

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a fundamental concept frequently encountered in algebra. It is expressed as: \[ y = mx + b \]where:

• \( m \) is the slope of the line, representing the steepness or incline of the line. It measures how much \( y \) increases or decreases for each unit change in \( x \).
• \( b \) is the y-intercept, which specifies the point where the line crosses the y-axis.

By rearranging this equation or manipulating the values of \( m \) and \( b \), you can gain insights into the graph's characteristics. Understanding the slope-intercept form helps in finding the angle of inclination, which forms the basis for more advanced concepts like trigonometric analyses of lines.

Inverse Tangent

Inverse tangent, also known as arctan or \( an^{-1} \), is an important trigonometric function used to determine an angle from its tangent value. In the context of finding the inclination of a line, if you have a slope \( m \), you can find the angle \( \theta \) of the line's inclination with the horizontal using:\[ \theta = \arctan(m) \]For example, with a slope \( m = \frac{3}{4} \), the calculation is:\[ \theta = \arctan \left( \frac{3}{4} \right) \]This function is vital because it allows translating a purely geometric concept (the slope) into an angle that can be used in trigonometric expressions. Calculators have built-in functions to quickly compute this, allowing efficient conversions from slope to angular measure.

Angle Conversion

Once the angle of inclination has been computed in radians using the inverse tangent function, converting this angle to degrees is often necessary. The standard conversion formula from radians to degrees is:\[ \text{degrees} = \text{radians} \times \frac{180}{\pi} \]Radians and degrees are two units of angular measurement:

• Radians relate directly to the circle's radius, often making it convenient for mathematical calculations.
• Degrees divide circles into 360 equal parts and are more intuitive for everyday use.

For example, if \( \theta_{radians} = \arctan \left( \frac{3}{4} \right) \), then\[ \theta_{degrees} = \theta_{radians} \times \frac{180}{\pi} \]This conversion is crucial in various fields such as engineering and physics, where angles may need to be expressed differently based on context or conventions.