Problem 19 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The inclination \(\theta\) of the line passing through the points \((\sqrt{3}, 2)\) and \((0, 1)\) is \(\frac{\pi}{6}\) radians or \(30°\).

Step 1: Calculation of slope

Calculate the slope using the formula: \(slope = \frac{y_2 - y_1}{x_2 - x_1}\). For points \((\sqrt{3}, 2)\) and \((0, 1)\), \(x_1 = \sqrt{3}\), \(y_1 = 2\), \(x_2 = 0\), and \(y_2 = 1\). So, the slope is calculated as \(slope = \frac{1 - 2}{0 - \sqrt{3}} = \frac{-1}{-\sqrt{3}} = \frac{\sqrt{3}}{3}\).

Step 2: Calculation of Inclination in Radians

Inclination of line in radians is given by \(\theta = tan^{-1} (slope)\). Substituting the value of slope from Step 1, \(\theta = tan^{-1}(\frac{\sqrt{3}}{3}) = \frac{\pi}{6}\).

Step 3: Conversion of Radians to Degrees

To convert radians into degrees, the formula is \(\theta_{degree} = \theta_{radian} \times \frac{180}{\pi}\). So, \(\theta_{degree} = \frac{\pi}{6} \times \frac{180}{\pi} = 30°\).

Key Concepts

Finding Line Inclination

The inclination of a line is a measure of the angle between the line and the positive direction of the x-axis. To find this angle, often denoted as \theta, we must understand how it relates to the slope of the line. To calculate the inclination, we need to know at least two points that lie on the line.

For example, let's take the points \(\sqrt{3}, 2\) and \(0, 1\) from our exercise. We can intuitively understand that the line passing through these points isn't horizontal (slope of 0) or vertical (undefined slope) and therefore, it would have a specific inclination to the x-axis. By finding the slope first, which dictates how steep the line is, we can then use trigonometric functions to find the angle that represents the line's inclination.

Slope of a Line

The slope of a line is essentially a ratio that describes how much the line rises ('the rise') for every unit it goes to the right ('the run'). This concept is a fundamental part of coordinate geometry and is symbolized as 'm'. To find the slope between two points, \(x_1, y_1\) and \(x_2, y_2\), we use the formula \[slope = \frac{y_2 - y_1}{x_2 - x_1}\].

In the given exercise, if we apply the formula to the points \(\sqrt{3}, 2\) and \(0, 1\), we obtain \[slope = \frac{1 - 2}{0 - \sqrt{3}}\], which simplifies to \(\frac{\sqrt{3}}{3}\). This value tells us that for every three units we move horizontally to the right, the line rises by \(\sqrt{3}\) units.

Radians to Degrees Conversion

Angles can be measured in various units, with radians and degrees being the most common. The radian is a measure of angle based on the radius of a circle, whereas the degree is a unit that divides one complete circle into 360 equal parts. The mathematical relationship between radians and degrees is crucial for converting one to the other.

To convert radians to degrees, we multiply the radian measure by \(\frac{180}{\pi}\). For instance, an inclination of \(\frac{\pi}{6}\) radians from our exercise would be converted to degrees by multiplying \(\frac{\pi}{6}\) by \(\frac{180}{\pi}\), resulting in 30 degrees. This conversion is vital when presenting angles in a more conventional format that is widely used in various applications outside of pure mathematics.