Problem 35 - Precalculus with Limits

Question

In Exercises 27-36, find the inclination \(\theta\) (in radians and degrees) of the line. \(5x + 3y = 0\)

Step-by-Step Solution

Answer

The inclination of the line \(5x + 3y = 0\) is approximately -1.03037683 radians or -59.0362435 degrees.

Step 1: Convert to Slope-Intercept Form

Firstly, we need to convert \(5x + 3y = 0\) to the form of \(y = mx + c\). For that, rewrite the equation, isolating 'y': \(-3y = 5x\) or \(y = -5/3 x\). In this case, m (the slope of the line) is -5/3.

Step 2: Find the Inclination in Radians

The angle of inclination, \(\theta\), of a line with slope 'm' can be calculated as \(\theta = arctan(m)\). Dominantly, arctan represents the inverse tangent. Here, we substitute our slope, -5/3, in this formula, to get \(\theta = arctan(-5/3) = -1.03037683\) radians.

Step 3: Convert Radians to Degrees

There is a known conversion from radians to degrees, that states 1 radian equals approximately 57.2958 degrees. Multiplying our result by this conversion factor, we get \(\theta = -1.03037683 * 57.2958 = -59.0362435\) degrees.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is fundamental in algebra and geometry to analyze the properties of a line in a coordinate system. The slope-intercept form is written as \( y = mx + c \), where \( m \) is the slope of the line and \( c \) is the y-intercept, which is the point where the line crosses the y-axis.

Consider the equation \( 5x + 3y = 0 \) from our exercise. We can change it to slope-intercept form by solving for \( y \) as follows: \( 3y = -5x \), and then \( y = -\frac{5}{3}x \). Here, \( -\frac{5}{3} \) represents the slope, indicating that for every three units we move horizontally, the vertical change is a decrease of five units. This allows for a graphical representation of the line and an understanding of its steepness and direction.

Angle of Inclination

The angle of inclination, denoted \( \theta \), is the angle that the line makes with the positive direction of the x-axis. It's a measure of the steepness, or incline, of a line. For a given slope \( m \), we find the angle of inclination using the arctangent function: \( \theta = \arctan(m) \).

Using our example with the slope \( -\frac{5}{3} \), the angle of inclination is \( \arctan(-\frac{5}{3}) \) which corresponds to a line sloping downwards from left to right. Remember that the arctangent function can handle all the slopes, providing angles in a range that covers all possible directions of lines. Hence, arctangent guarantees we always get the principal value of \( \theta \), which represents the smallest positive angle that can describe the line's incline.

Radians to Degrees Conversion

Radians and degrees are two units for measuring angles. They can be inter-converted using the conversion factor where \( 1 \) radian is approximately \( 57.2958 \) degrees. To convert radians to degrees, multiply the angle in radians by this conversion factor.

For example, if we have an angle of \( -1.03037683 \) radians (from our angle of inclination calculation), to find the degrees, we use the conversion as follows: \( -1.03037683 \) radians \( \times 57.2958 \approx -59.0362435 \) degrees. Hence, the negative sign indicates the angle measures in the clockwise direction from the positive x-axis. It's essential to grasp this concept to accurately interpret and translate the angular measurements between systems used in different applications and fields of study.