Question

Angle of Inclination Find the slope of the line having the given angle of inclination. $$38.2^{\circ}$$

Short answer

The slope of the line is the tangent of 38.2 degrees, which is approximately 0.78.

Step-by-step solution

Understanding 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 can be found using the tangent of the angle of inclination.

Calculating the Slope

Calculate the slope (m) of the line using the tangent of the given angle. The formula to calculate the slope is m = \( \tan(\theta) \), where \( \theta \) is the angle of inclination. For an angle of inclination of 38.2 degrees: m = \( \tan(38.2^\circ) \).

Using a Calculator

Use a scientific calculator to find the value of \( \tan(38.2^\circ) \). Make sure the calculator is set to degree mode before calculating the tangent of the angle.

Key concepts

Slope Calculation

To understand slope calculation, consider a line on a Cartesian coordinate system. It tells us how steep the line is, and it's a measure of vertical change relative to the horizontal change between any two points on the line. Mathematically, slope is typically represented as 'm' and is calculated as the ratio of the rise (the vertical change) over the run (the horizontal change).

When it comes to the angle of inclination, the slope is directly related to this angle. The angle of inclination is the angle formed when a straight line tilts away from the positive x-axis, and it's measured in degrees. To find the slope from this angle, we use the trigonometric function 'tangent'. Essentially, the slope is the tangent of the angle of inclination, expressed as:
m = \tan(\theta),
where \(\theta\) is the angle the line makes with the positive x-axis. For example, a line with an inclination of 38.2 degrees has a slope that can be calculated using this precise method.

It's also important to note that slope can be positive, negative, zero, or undefined. Positive slope indicates a line rising as you move from left to right, while a negative slope indicates a line falling. A zero slope means the line is horizontal, and an undefined slope indicates a vertical line.

Tangent of Angle

The tangent of an angle is a fundamental concept in trigonometry, which relates to the angle's slope on a graph. When a line is graphed, and an angle of inclination is formed with the x-axis, the tangent of that angle gives the line's slope. You can remember this relationship by recalling that the tangent of an angle in a right triangle is the ratio of the side opposite to the angle to the side adjacent to it.

The mathematical representation for the tangent function is written as \tan(\theta), where \(\theta\) is the angle. For the given exercise, finding the slope of a line with a 38.2-degree inclination involves computing \tan(38.2^\circ). This computation will yield a numerical value that represents the slope of the line. It's beneficial to understand that the tangent function will repeat its values every 180 degrees due to periodicity, which means that the slope for angles that differ by 180 degrees will be the same.

Using a Scientific Calculator

A scientific calculator is an invaluable tool for students, particularly when computing trigonometric functions like the tangent. To use it for finding the tangent of an angle, you should first ensure that the calculator is in the correct mode—degrees or radians—based on the unit of the angle given. Since our exercise uses degrees, we must set it to degree mode.

To find the tangent, you'll typically enter the angle (say, 38.2) and then press the 'TAN' button. Here are the steps you might follow:

• Turn on your calculator and check if it's in degree mode (it might indicate this with a 'DEG' symbol on the screen).
• Type in the angle of inclination (in this case, 38.2).
• Press the 'TAN' key to get the tangent of this angle, which gives you the slope of the line.

Be mindful that the process might slightly vary depending on the calculator brand and model. If your result seems odd or unexpected, check that the calculator is in the correct mode and that you haven't made a keying error. Scientific calculators are designed to handle numerous functions and often have dedicated keys for trigonometric functions, making slope calculation a breeze once you become familiar with the tool.