Problem 33 - Precalculus with Limits

Question

In Exercises 27-36, find the inclination \(\theta\) (in radians and degrees) of the line. \(6x - 2y + 8 = 0\)

Step-by-Step Solution

Answer

The inclination of the line \(6x - 2y + 8 = 0\) is approximately 1.249 radians or 71.57 degrees.

Step 1: Identify A and B from the equation

From the equation \(6x - 2y + 8 = 0\), A is equivalent to 6 and B is equivalent to -2.

Step 2: Calculate the slope

The slope of the line (m) can be found as \(-A/B\). Substituting the values of A and B gives \(m = -6/-2 = 3\)

Step 3: Find the inclination in radians

Inclination in radians, \(\theta\), is given by \(arctan(m)\). Substituting m=3 gives \(arctan(3)\) which is approximately 1.249 radians.

Step 4: Find the inclination in degrees

To convert radians to degrees, multiply the radian value by \(180/\pi\). So the inclination in degrees is \(1.249 * 180/\pi\) which is approximately 71.57 degrees.

Key Concepts

Slope of a Line

Understanding the slope of a line is critical when learning algebra and geometry. The slope is a measure of how steep a line is. It is usually represented by the letter 'm' and is calculated as the 'rise over run,' which means the vertical change divided by the horizontal change between two points on the line. To find the slope of a line given by the equation in the form of ax + by = c, one can use the formula m = -a/b.

For example, in our exercise equation 6x - 2y + 8 = 0, identifying a as 6 and b as -2, we substitute them into our formula to get m = -6/-2, which simplifies to m = 3. Thus, the slope of the line is 3, indicating that for every unit increase in x, y increases by 3 units. This value not only tells us the inclination of the line as an abstract number but now serves as a significant input to find the angle of inclination.

Radians to Degrees Conversion

Angle measurement can be in degrees or radians—the two major units used in mathematics. While degrees are often used in daily life and earlier education, radians are the standard unit of angular measure used in advanced mathematics and sciences. To convert radians to degrees, one can use the formula degree = radian * (180/π).

In our exercise, after finding the inclination in radians to be approximately 1.249, we need to convert this value into degrees. Applying our conversion formula, we get 1.249 * (180/π) which equals approximately 71.57 degrees. Understanding this conversion is essential when working with trigonometric functions and interpreting angles for practical applications, like navigation or engineering projects.

Arctan Function

The arctan function, also known as the inverse tangent function, is used to find the angle whose tangent is a given number. It's symbolized as arctan or tan^-1. This function is crucial for converting the slope of a line into the angle of inclination, which can be more understandable and visually relatable.

In the context of our exercise, after determining the slope of the line to be 3, we can find the inclination angle in radians by calculating arctan(3). This gives us an angle in radians, which we can then easily convert to degrees if needed. It's important to remember that while the slope provides us with a ratio, the arctan function translates that ratio into an actual angle, which describes the steepness in terms of degrees or radians.