Problem 31 - Precalculus with Limits

Question

In Exercises 27-36, find the inclination \(\theta\) (in radians and degrees) of the line. \(x + \sqrt{3}y + 2 = 0\)

Step-by-Step Solution

Answer

\(\theta\) is \(-\pi/6\) radians or \(-30\) degrees.

Step 1: Find the inclination in radians

To find the inclination in radians, utilize the formula \(\theta = \arctan(-a/b)\). Here \(a = 1\) and \(b = \sqrt{3}\). Subtracting these values, we get \(\theta = \arctan(-1/\sqrt{3})\). Solving this, we get \(\theta = -\pi/6\) radians.

Step 2: Convert the inclination from radians to degrees

To convert \(\theta\) from radians to degrees, multiple by the conversion factor \(\frac{180}{\pi}\). Here, it goes like \(\theta = -\pi/6 * \frac{180}{\pi}\). Solving, we get \(\theta = -30\) degrees.

Key Concepts

Understanding the Arctan Function

The arctan (or inverse tangent) function is essential in finding the angle of inclination of a line. If you're given a line equation in the form of \(ax + by + c = 0\), the slope \(m\) is \(-\frac{a}{b}\).

The function \(\arctan(m)\) is used to find the angle \(\theta\) in radians. In our exercise, we had \(a = 1\) and \(b = \sqrt{3}\), leading to a slope of \(-\frac{1}{\sqrt{3}}\). Thus, the inclination \(\theta\) is given by \(\theta = \arctan\left(-\frac{1}{\sqrt{3}}\right)\).

• Why radians? Radians provide a direct measurement of angles based on the unit circle, making them a natural fit for calculus and trigonometry.
• Significance: The result \(-\frac{\pi}{6}\) radians shows the direction and steepness of the line.

Converting Radians to Degrees

Once you've determined an angle in radians, it's often helpful to express it in degrees for intuitive understanding. Degrees are more commonly used in everyday life and various disciplines.

To convert radians to degrees, use the conversion factor \(\frac{180}{\pi}\). Multiplying the radian measure by this factor allows you to express the angle in degrees.

• For our problem: \(-\frac{\pi}{6} \times \frac{180}{\pi} = -30\) degrees.
• This conversion helps relate the mathematical result to familiar angles like 30°, 45°, and 90°.

Conversions between radians and degrees allow versatility in interpreting angles. It helps communicate more effectively, especially when both scientific and everyday contexts are involved.

Exploring Linear Equations

Linear equations are fundamental in algebra and geometry, representing straight lines on a graph. The general form \(ax + by + c = 0\) allows us to derive key properties of the line, like slope and inclination.

The key steps involve:

• Identifying Slope: Given the equation \(x + \sqrt{3}y + 2 = 0\), we can rearrange it to find the slope: \(y = -\frac{1}{\sqrt{3}}x - \frac{2}{\sqrt{3}}\).
• Using Slope to Find Inclination: The slope \(m = -\frac{1}{\sqrt{3}}\) points to how steep the line is compared to the x-axis.

Linear equations form the backbone of many mathematical concepts and applications. Understanding them helps in solving diverse problems involving intersection points, parallel lines, and more.