Problem 21 - Precalculus with Limits

Question

In Exercises 19-26, find the inclination \(\theta\) (in radians and degrees) of the line passing through the points. \((-\sqrt{3}, -1)\), \((0, -2)\)

Step-by-Step Solution

Answer

The inclination of the line passing through the points \((- \sqrt{3}, -1)\) and \((0, -2)\) is approximately \(1.047\) radians or \(60\) degrees.

Step 1: Identify the correct points

First, identify the coordinate points given which are \((- \sqrt{3}, -1)\) and \((0, -2)\). Label them as point 1 (\(x_1, y_1\)) and point 2 (\(x_2, y_2\)) respectively. So, \(x_1 = - \sqrt{3}\), \(y_1 = -1\), \(x_2 = 0\), and \(y_2 = -2\).

Step 2: Calculate the slope

Next, use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) to calculate the slope or inclination of the line. Plug the values \(y_2 = -2\), \(y_1 = -1\), \(x_2 = 0\), and \(x_1 = -\sqrt{3}\) into the formula to get: \( m = \frac{-2 - (-1)}{0 - (-\sqrt{3})} = \sqrt{3}\).

Step 3: Find the inclination in radians

The inclination of the line in radians is given by the arctangent of the slope. Using a calculator, calculate \( \arctan{(m)} \). In this case, \(\arctan{(\sqrt{3})} \approx 1.047 \) radians.

Step 4: Convert the inclination to degrees

Finally, convert the radians into degrees by multiplying the radian result by \(\frac{180}{\pi}\). The conversion will yield approximately \(60\) degrees.

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep the line is. It's calculated using two points on the line. This is often represented by the letter \(m\). To find the slope, you use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. The numerator \((y_2 - y_1)\) represents the vertical change, while the denominator \((x_2 - x_1)\) represents the horizontal change.

In this exercise, substituting the given points \((- \sqrt{3}, -1)\) and \((0, -2)\) into the formula, you find:

• \( m = \frac{-2 - (-1)}{0 - (-\sqrt{3})} = \sqrt{3} \)

This tells us that for each unit increase in \(x\), \(y\) decreases by \(\sqrt{3}\) units.

Arctangent Function

The arctangent function, written as \(\arctan\), is the inverse of the tangent function. It helps to find the angle whose tangent is a given number. When you know the slope \(m\) of a line, the arctangent function helps find the angle \(\theta\) the line makes with the x-axis.

Using the slope \(m = \sqrt{3}\), the formula is:

• \( \theta = \arctan(m) \)

Calculating this:

• \( \theta = \arctan(\sqrt{3}) \approx 1.047 \text{ radians} \)

This angle is the inclination of the line in radians, showing how the line tilts with respect to the horizontal.

Conversion Between Radians and Degrees

Radians and degrees are two units of measuring angles. Converting between them is crucial, especially when solving problems that use both units. The conversion factor is \(\frac{180}{\pi}\). To convert radians to degrees, you multiply the radian measure by this factor.

Given that the inclination \(\theta\) is \(1.047\) radians, the conversion is:

• \( \theta_{\text{degrees}} = 1.047 \times \frac{180}{\pi} \approx 60 \text{ degrees} \)

This conversion helps relate the angle in a more commonly used unit of measurement, making it easier to understand and apply in various contexts.