Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the figure, which shows two lines whose equations are \(y_{1}=m_{1} x+b_{1}\) and \(y_{2}=m_{2} x+b_{2}\). Assume that both lines have positive slopes. Derive a formula for the angle between the two lines. Then use your formula to find the angle between the given pair of lines. $$\begin{aligned} &y=x\\\ &y=\frac{1}{\sqrt{3}} x \end{aligned}$$

Step-by-Step Solution

Answer

The angle between the lines \(y=x\) and \(y=1/\sqrt{3}x\) is approximately \(\theta = \arctan(\frac{\sqrt{3}-3}{1+\sqrt{3}}) * 180/ \pi\) degrees.

Step 1: Derive the formula for the angle

The formula for the angle θ between two lines with slopes \(m_1\) and \(m_2\) can be derived from the tangent of the angle. The formula is \[\tan(\theta) = \frac{m_2 - m_1}{1+m_1*m_2}\]It is based on the geometric fact that the tangent of the angle between two intersecting lines is equal to the difference of their slopes divided by one plus the product of their slopes.

Step 2: Identify the slopes of the given lines

In the equation of a line in the form \(y = mx+b\), the value of \(m\) is the slope. The first line \(y=x\) has a slope of 1, and the second line \(y=1\sqrt{3}x\) has a slope of \(1\sqrt{3}\) .

Step 3: Calculate the angle

Substitute the slopes into the formula.So, \[\tan(\theta) = \frac{1/\sqrt{3} - 1}{1+(1*1/\sqrt{3})} = \frac{\sqrt{3}-3}{1+\sqrt{3}}\]To find the angle θ, we then use the inverse tangent function (also known as the arctangent), which is often represented as 'atan' or 'tan^{-1}' on calculators. Because we want the angle in degrees, we multiply the result by \(180/\pi\) to convert from radians to degrees. Therefore, \[\theta = \arctan(\frac{\sqrt{3}-3}{1+\sqrt{3}}) * 180/ \pi\]Calculate the arctan(), then multiply the result by \(180/\pi\) to get the final answer.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in geometry and calculus. It tells us how steep the line is and in which direction it inclines. In the equation of a line, usually written as \(y = mx + b\), the letter \(m\) represents the slope.
The slope \(m\) is calculated as the change in the vertical direction (rise) divided by the change in the horizontal direction (run). This can be expressed as:

• \( m = \frac{\Delta y}{\Delta x} \)

The slope is positive when the line ascends from left to right, and negative if it descends. If \( m = 0 \), the line is horizontal, while a vertical line has an undefined slope.
Understanding the slope is crucial for determining the angle between lines, as the slope helps define the tangent of the angle between two lines.

Tangent Function

The tangent function \(\tan(\theta)\) is a trigonometric function that relates an angle of a right triangle to the ratio of the opposite side over the adjacent side. It is one of the basic functions used extensively in trigonometry.
This function is crucial when calculating the angle \( \theta \) between two lines, as the tangent of that angle is related to the slopes of the lines. For two lines with slopes \( m_1 \) and \( m_2 \), the formula for the tangent of the angle \( \theta \) is:

• \( \tan(\theta) = \frac{m_2 - m_1}{1 + m_1 * m_2} \)

This equation is derived from the property of tangents, and shows that the angle between two lines is influenced by their respective slopes. Thus, knowing the tangent function allows us to find the angle, laying the groundwork for using the arctangent function to determine the exact size of the angle in a practical setting.

Arctangent

The arctangent function, sometimes called inverse tangent or \(\tan^{-1}\), is used to find an angle when its tangent value is known. It is essentially the reverse operation of the tangent function.
In our context, after finding the value of \(\tan(\theta)\) using the formula \(\frac{m_2 - m_1}{1 + m_1 * m_2}\), we apply the arctangent function to determine the angle \(\theta\) itself. Sing larger calculators or computers, this function is usually provided as \(atan\) or \(tan^{-1}\).
When we compute \(\theta = \arctan(\tan(\theta))\), the result is given in radians. To convert from radians to degrees, which is often desired for practical purposes, each radian is multiplied by \(\frac{180}{\pi}\). Thus, the full process involves calculating the arctan of the value from our earlier slope-derived formula, ensuring we understand and correctly quantify the relationship between two intersecting lines.