Problem 28 - Precalculus with Limits

Question

In Exercises 27-36, find the inclination \(\theta\) (in radians and degrees) of the line. \(x - \sqrt{3}y + 1 = 0\)

Step-by-Step Solution

Answer

The inclination of the line \(x - \sqrt{3}y + 1 = 0\) is \(π/3\) radians or 60 degrees.

Step 1: Determine the slope of the line

The slope (m) of the line from its equation \(x - \sqrt{3}y + 1 = 0\) can be found as m = -(-1/\sqrt{3}) = \sqrt{3}

Step 2: Find the inclination angle in radians

The inclination angle (θ) in radians is the arctan of the slope, so θ = arctan(m) = arctan(\sqrt{3}) , which is θ = π/3 radians.

Step 3: Convert the angle to degrees

The angle in degrees can be found by multiplying the radian value by \((180/π)\). So, θ = π/3 * (180/π) = 60 degrees.

Key Concepts

Slope of a Line

When we talk about the slope of a line, we're referring to a number that represents the steepness and direction of the line. The slope, often represented by the letter "m," indicates how much the line rises or falls as it moves from left to right.

• If the slope is positive, the line rises as it moves rightward.
• If the slope is negative, the line falls as it moves rightward.
• If the slope is zero, the line is horizontal.
• If the slope is undefined, the line is vertical.

The slope formula from a line equation in standard form, such as \(ax + by + c = 0\), is \(m = -\frac{a}{b}\). For the given line \(x - \sqrt{3}y + 1 = 0\), this results in a slope \(m = \sqrt{3}\). This tells us that for a one-unit move horizontally, the line rises \(\sqrt{3}\) units.

Radian to Degree Conversion

Converting between radians and degrees is an essential skill in trigonometry. Radians and degrees are both units used to measure angles. While degrees are perhaps more familiar, radians are crucial in calculus and many areas of mathematics because they make formula computations easier.

• A full circle is \(360\) degrees or \(2\pi\) radians.
• To convert from radians to degrees, you multiply by \(\frac{180}{\pi}\).
• To convert from degrees to radians, you multiply by \(\frac{\pi}{180}\).

In this exercise, we need to convert \(\pi/3\) radians to degrees. Using our conversion factor, \(\pi/3\times \frac{180}{\pi}\), cancels out \(\pi\) and simplifies to \(60\) degrees. Thus, the inclination in degrees is \(60\).

Arctan Function

The arctan function, also known as the inverse tangent function, is important when you want to find the angle corresponding to a particular slope. In trigonometric terms, if you know \(\tan(\theta)\), the arctan function helps you find \(\theta\).

• The function is represented as \(\arctan(x)\).
• It's used to calculate the angle whose tangent is a given number.
• Many calculators have an arctan button for this purpose.

In our exercise, we were given a slope \(m = \sqrt{3}\). By applying the \(\arctan\) function, we find the angle \(\theta = \arctan(\sqrt{3})\). This angle turns out to be \(\pi/3\) radians or \(60\) degrees, giving the desired inclination of the line in both measures.