Problem 32 - Technical Mathematics with Calculus

Question

Angle of Inclination Find the slope of the line having the given angle of inclination. $$156.3^{\circ}$$

Step-by-Step Solution

Answer

The slope of the line with an angle of inclination of 156.3 degrees is approximately -5.671.

Step 1: Define Angle of Inclination

The angle of inclination of a line is the angle formed by the intersection of the line and the positive x-axis. The slope of the line is the tangent of the angle of inclination.

Step 2: Convert the angle to radians

To find the slope using the tangent function, first convert the angle from degrees to radians. The conversion formula is radians = degrees * (\(\frac{\pi}{180}\)).

Step 3: Calculate the slope

Use the tangent function with the angle in radians to find the slope (\(m\)) of the line. The slope of the line is \(m = \tan(\text{angle in radians})\).

Key Concepts

Slope of a Line

Understanding the slope of a line is fundamental in algebra and geometry. The slope is a measure of how steep a line is. In more technical terms, it represents the ratio of the rise (vertical change) over the run (horizontal change) between any two points on the line. When we talk about the slope in the context of an angle of inclination, we're discussing a straight line's steepness as it tilts away from the horizontal axis.

The angle of inclination is the angle formed where the line intersects the positive x-axis, with the angle being measured counterclockwise from that axis. To find the slope from an angle of inclination, you use the tangent function, because the tangent of an angle in a right triangle is the ratio of the opposite side (rise) to the adjacent side (run). This relationship gives us a direct method to calculate a line's slope by using its angle of inclination.

Tangent Function

The tangent function is one of the three primary trigonometric functions, along with sine and cosine. It's defined in the context of a right-angled triangle as the ratio of the length of the opposite side to the length of the adjacent side. However, in the unit circle, which is often more helpful for understanding angles beyond 90 degrees, tangent represents the ratio of the y-coordinate to the x-coordinate of a point on the circle's circumference.

In terms of graphs, the tangent function produces a wave that oscillates between positive and negative infinity, crossing through zero at every multiple of \(\frac{\text{\textpi}}{2}\text{ radians}\text{ (90 degrees)}\). Using this function to determine the slope of a line allows us to handle steepness in terms of angles, connecting the geometric interpretation with algebraic calculations.

Radians Conversion

Radians are a unit of angular measurement used in mathematics to measure angles in terms of the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians provide a measurement based on the circle's radius. One full revolution of a circle is \(2\text\textpi\) radians, which is equivalent to 360 degrees.

To convert an angle from degrees to radians, you multiply the number of degrees by \(\frac{\text\textpi}{180}\text{ radians/degree}\text{)\). Thus, to convert the given 156.3 degrees into radians, the calculation would be \(156.3 \times \frac{\text\textpi}{180}\text{ radians/degree}\text{)\), which is essential when using the tangent function to find the slope of a line. Remember that converting to radians is a crucial step in many trigonometric problems involving circles and angles, as most trigonometric functions in calculators require inputs in radians.