Problem 93 - 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. $$\begin{aligned} &y=x\\\ &y=\sqrt{3} x \end{aligned}$$

Step-by-Step Solution

Answer

The angle between the two lines is 150 degrees.

Step 1: Calculate the Slopes

In this case, the equations given are y = x and y = \( \sqrt{3} \)x. From these equations, we can see that slopes m1 and m2 are 1 and \( \sqrt{3} \) respectively.

Step 2: Calculate the Angle

Substitute these values into the formula. We use the formula for finding the angle between two lines: \( \tan(\theta) = \frac{m1 - m2}{1 + m1*m2} \). So \( \tan(\theta) = \frac{1 - \sqrt{3}}{1 + 1*\sqrt{3}} = -1/\sqrt{3} \). Solving for theta gives us \( \theta = \tan^-1(-1/\sqrt{3}) \). So the angle between these lines is \( \theta = \tan^-1(-1/\sqrt{3}) \), or 150 degrees.

Key Concepts

Positive Slopes

In the realm of coordinate geometry, the slope of a line indicates its steepness and direction. When dealing with positive slopes, this signifies that the line rises as it moves from left to right.

• A positive slope means that the change in y is positive when x increases.
• Visually, lines with positive slopes will slant upwards to the right on a graph.

This is essential when determining the angle between two lines, as both lines having positive slopes means they both slant upwards, yet they can intersect at different angles depending on their steepness.For instance, in our exercise, we have two lines: one with a slope of 1 (\( m_1 = 1 \) ) and the other with a slope of \( \sqrt{3} \) ( \( m_2 = \sqrt{3} \). This sets up a situation where both lines rise, but at different rates due to their varying slopes.

Equations of Lines

The equation of a line in its simplest form is expressed as \( y = mx + b \), where:

• \( m \) is the slope: this tells us how tilted the line is.
• \( b \) is the y-intercept: this is where the line crosses the y-axis.

In problems like these, knowing the equation is crucial because it directly gives us the slope values needed to calculate the angle between lines.Take our given lines:- Line 1 is \( y = x \), which simplifies to a slope \( m_1 = 1 \) and implies \( b = 0 \).- Line 2 is \( y = \sqrt{3} x \), with a slope \( m_2 = \sqrt{3} \).These equations provide us with the necessary information to find where and how these lines intersect by comparing their slopes.

Tangent Formula

The tangent formula is a key tool in finding the angle between two intersecting lines. When you know the slopes of two lines, say \( m_1 \) and \( m_2 \), the angle \( \theta \) between them can be found using the formula:\[ \tan(\theta) = \frac{m_1 - m_2}{1 + m_1 \cdot m_2}\]This formula provides us with the tangent of the angle. You can then use the inverse tangent function, usually written as \( \tan^{-1} \), to find the angle \( \theta \).Applying this to our specific exercise:- Substituting the slopes, \( m_1 = 1 \) and \( m_2 = \sqrt{3} \), into the formula gives:\[ \tan(\theta) = \frac{1 - \sqrt{3}}{1 + \sqrt{3}}\]- This simplifies to \( \tan(\theta) = -\frac{1}{\sqrt{3}} \).Taking \( \tan^{-1}(-\frac{1}{\sqrt{3}}) \) gives us \( \theta = 150 \) degrees, which is the angle between the two lines. This method is efficient and relies on standard trigonometric operations.Remember, knowing how to manipulate these formulas allows you to solve for angles between lines quickly and accurately.