Question

Angle of Inclination Find the slope of the line having the given angle of inclination. $$58^{\circ} 14^{\prime}$$

Short answer

To find the slope for an angle of inclination of 58 degrees and 14 minutes, convert the minutes to decimal form, calculate the tangent of the resulting angle, and the result will be the slope.

Step-by-step solution

Convert angle to decimal degrees

Firstly, convert the angle of inclination from degrees and minutes into decimal degrees. There are 60 minutes in a degree, so you divide the number of minutes by 60 to convert them into a fraction of a degree. Finally, add this value to the number of whole degrees to get the angle in decimal form. For an angle of inclination of 58 degrees and 14 minutes, the conversion to decimal degrees is: \(58 + \frac{14}{60} = 58 + 0.2333 = 58.2333\) degrees.

Find the slope using the tangent

The slope of the line is the tangent of the angle of inclination. Use a scientific calculator to find the tangent of the angle, making sure the calculator is set to degrees mode. Slope (m) = \(\tan(\text{angle of inclination})\)Slope (m) = \(\tan(58.2333^\circ)\)

Calculate the slope

Using the scientific calculator, calculate the tangent of 58.2333 degrees to find the slope of the line. The slope will be the numerical value that you get as an output.

Key concepts

Slope Calculation

When we talk about the slope of a line in mathematics, we're referring to a measure of the steepness or incline of that line. Often represented by the letter 'm', slope is a crucial concept in algebra and geometry.

To calculate the slope of a line given its angle of inclination, we use the trigonometric function known as the tangent. The tangent of an angle in a right triangle is the ratio of the side opposite to the angle (rise) to the side adjacent to the angle (run). In a coordinate plane, this correlates to the vertical change divided by the horizontal change between two points on a line.

Calculating the Slope from an Angle

Imagine you have a line that creates an angle with the positive x-axis, which is known as the angle of inclination. The tangent of this angle gives us the slope of the line. In other words, if the angle of inclination is represented by 'θ', then we can say that the slope of the line, 'm', is equal to \( \tan(θ) \). This is because the tangent of the angle essentially describes how much the line rises vertically for every unit of horizontal movement. If the angle of inclination increases, the slope becomes steeper, and the tangent value grows larger.

Tangent of an Angle

The tangent function is one of the most important trigonometric functions, especially when dealing with right-angled triangles and the angles within them. It is often introduced after sine and cosine and completes the primary set of trigonometric functions used in various calculations.

In a right-angled triangle, the tangent of one of the non-right angles, let's label it as 'θ', is found by dividing the length of the opposite side by the length of the adjacent side. Algebraically, it is expressed as:\( \tan(θ) = \frac{\text{opposite}}{\text{adjacent}} \).

Using Tangent in Slope Calculation

When it comes to finding the slope of a line, the angle of inclination and the tangent function come into play together. By taking the tangent of the angle of inclination, we obtain the slope. This is because within the context of coordinate planes and lines, the tangent ratio expresses the same rise-over-run relationship that defines the slope of a line.

Decimal Degree Conversion

Angles can be expressed in various formats, and decimal degrees are one form that is especially useful in mathematical calculations. When an angle is given in degrees, minutes, and seconds, converting it to decimal degrees can simplify many types of computations, such as calculating a line's slope or using trigonometric functions.

One degree is equal to 60 minutes (') and one minute is equal to 60 seconds (''). To convert an angle given in degrees and minutes to decimal degrees, we follow a simple two-step process:

• Divide the number of minutes by 60 to convert them to a decimal.
• Add the decimal to the whole number of degrees.

For example, if we have an angle of \(58^\circ 14'\), we convert it by dividing 14 by 60, yielding approximately 0.2333. Adding this to the 58 degrees, we get the angle in decimal degrees: \(58 + 0.2333 = 58.2333^\circ\).

This conversion plays a key role in simplifying the calculations whenever we work with angles, as most scientific calculators require angles in decimal degrees when working with trigonometric functions.