Question

Angle of Inclination Find the slope of the line having the given angle of inclination. $$132.8^{\circ}$$

Short answer

The slope of the line with an angle of inclination of 132.8 degrees is approximately -1.082.

Step-by-step solution

Understand Angle of Inclination

The angle of inclination of a line is the angle measured from the x-axis to the line in a counter-clockwise direction. The slope of the line (m) can be found using the tangent of the angle of inclination, i.e., \( m = \tan(\theta) \), where \( \theta \) is the angle of inclination.

Convert Angles Greater Than 90 Degrees

When the angle of inclination is greater than 90 degrees, the line will have a negative slope. We have to adjust the given angle by subtracting it from 180 degrees to find the acute angle that will give us the negative slope. For our angle, \( 132.8^\circ \), the calculation would be \( 180^\circ - 132.8^\circ = 47.2^\circ \).

Calculate the Slope Using Tangent

Find the tangent of the acute angle (47.2 degrees) to determine the slope of the line. In this case, \( m = \tan(47.2^\circ) \).

Determine the Sign of the Slope

Since the original angle of inclination is greater than 90 degrees, the slope will be negative. Thus, we must attach a negative sign to our result from step 3.

Utilize a Calculator

Utilize a calculator to find \( \tan(47.2^\circ) \) and attach the negative sign. The result is the slope of the line.

Key concepts

Angle of Inclination

Understanding the angle of inclination is crucial when you're studying linear functions and their graphs. This angle is essentially the measure of how much a line tilts away from the horizontal axis, which is the x-axis in the coordinate system. It's defined as the angle made by the line and the positive direction of the x-axis, measured counter-clockwise. The larger the angle of inclination, the steeper the line appears.

One might wonder why angles greater than 90 degrees result in negative slopes. When you have a line that's inclined more than 90 degrees, it actually means the line is sloping downwards as it moves from left to right, which is why it gets a negative value. Therefore, every time you are given an angle greater than 90 degrees, like the 132.8 degrees in our problem, you'll know right away that the line has a downward trend, indicating a negative slope.

Tangent Function

The tangent function comes into play when working with angles and slopes. It's one of the trigonometric functions which relates an angle in a right-angled triangle to the ratio of its opposite side over its adjacent side. In the context of linear equations, the tangent of an angle of inclination gives us the slope of the line, denoted as 'm'.

This makes perfect sense when you imagine a right triangle formed by the line, the x-axis, and the vertical line passing through the point where the line intersects the x-axis.

Practical Application

As for how to apply this in a problem, you'd take the tangent of the angle of inclination as it relates to a horizontal line. But if the given angle exceeds 90 degrees, like the 132.8-degree angle in the exercise, you need to find the equivalent acute angle by subtracting it from 180 degrees, which relates to a line with a negative slope going downwards as it goes to the right.

Negative Slope

Let's dive a bit deeper into the concept of a negative slope, as it is a fundamental characteristic of certain line graphs. A negative slope indicates that the line is descending from left to right. This contrasts with a positive slope, where the line rises from left to right. So, in essence, the 'negative' part is like a direction signal, telling us that the line is decreasing in value.

The significance of a negative slope is quite vivid in almost any graph that's depicting decline, like falling temperatures over time or decreasing prices in a supply and demand curve.

Using the tangent function to calculate a negative slope follows the same principles as with a positive slope, but after finding the value using a calculator or a table, you affix a negative sign to denote the line's downward trajectory in the coordinate plane. This simple sign tells a whole story about the direction and behavior of a line on a graph.