Problem 32 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line passing through the points. $$(0,100),(50,0)$$

Step-by-Step Solution

Answer

The inclination \(\theta\) of the line is \(\arctan(-2)\) radians or -114.59 degrees.

Step 1: Find the slope of the line

Use the formula \((y_2 - y_1) / (x_2 - x_1)\) to calculate the slope \(m\) of the line passing through the two points \((0,100)\) and \((50,0)\). This gives \(m = (0 - 100) / (50 - 0) = -2\).

Step 2: Find the inclination of the line

The inclination, or angle \(\theta\), of the line is the angle that the line makes with the positive x-axis. We can find this value by using \(\arctan\) of the slope, this gives \(\theta = \arctan(m) = \arctan(-2)\) radians.

Step 3: Convert the inclination from radians to degrees

We need to convert \(\theta = \arctan(-2)\) from radians to degrees to answer part of the question. The conversion is calculated using the formula \(degrees = radians \times (180 / \pi)\). Substitute the given values to find the value in degrees: \(-2 \times (180 / \pi) = -114.59 degrees\).

Key Concepts

Slope of a Line

Understanding the slope of a line is critical in many areas of mathematics and science. It represents how steep a line is and is a measure of the line's inclination. In simpler terms, the slope is a ratio of the vertical change, or the rise, to the horizontal change, or the run, between two points on a line.

When we have two points, \( (x_1, y_1) \) and \( (x_2, y_2) \) on a Cartesian plane, the slope \(m\) is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In the given exercise, the points are \( (0, 100) \) and \( (50, 0) \). So, \(m = \frac{0 - 100}{50 - 0} = -2\). A negative slope indicates that the line is descending as it moves from left to right.

Trigonometric Ratios

Trigonometric ratios provide a connection between the angles and sides of a right triangle. They are fundamental in linking the concepts of geometry with those of trigonometry. The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).

For an angle \(\theta\), these ratios can define

• The sine of \(\theta\) (\text{sin}\theta) is the ratio of the length of the opposite side to the hypotenuse.
• The cosine of \(\theta\) (\text{cos}\theta) is the ratio of the adjacent side to the hypotenuse.
• The tangent of \(\theta\) (\text{tan}\theta) is the ratio of the opposite side to the adjacent side, and it's particularly relevant when discussing the slope of a line.

Using the tangent ratio, we can find the inclination angle of a line by applying \(\theta = \arctan(m)\), where \(m\) is the slope of the line.

Converting Radians to Degrees

Radians and degrees are two units of measure for angles. Since they measure the same thing, we can convert from one to the other. Radians are often used in higher mathematics because they provide a more direct measurement of angle related to the arc length of a circle.

To convert radians to degrees, we use the formula \(\text{degrees} = \text{radians} \times \frac{180}{\pi}\). Why 180 divided by \(\pi\)? Because \(\pi\) radians is equal to 180 degrees, which corresponds to half of the circle's circumference.

Applying the Conversion

In our example, we first found the angle \(\theta\) in radians as \(\arctan(-2)\). To convert this value to degrees, we multiply by \(\frac{180}{\pi}\), as shown in the solution. The negative sign indicates that the angle is measured in the clockwise direction from the positive x-axis.