Problem 64 - Algebra and Trigonometry

Question

(a) If \(L\) is a line in the plane and \(\theta\) is the angle formed by the line and the \(x\) -axis as shown in the figure, show that the slope \(m\) of the line is given by $$ m=\tan \theta $$ (b) Let \(L_{1}\) and \(L_{2}\) be two nonparallel lines in the plane with slopes \(m_{1}\) and \(m_{2},\) respectively. Let \(\psi\) be the acute angle formed by the two lines (see the following figure). Show that $$ \tan \psi=\frac{m_{2}-m_{1}}{1+m_{1} m_{2}} $$ (c) Find the acute angle formed by the two lines $$ \begin{aligned} y=\frac{1}{3} X+1 & \text { and } \quad y=-\frac{1}{2} x-3 \end{aligned} $$ (d) Show that if two lines are perpendicular, then the slope of one is the negative reciprocal of the slope of the other. [Hint: First find an expression for \(\cot \psi . ]\)

Step-by-Step Solution

Answer

(a) \(m = \tan \theta\). (b) \(\tan \psi = \frac{m_2 - m_1}{1 + m_1 m_2}\). (c) \(\psi \approx 24.44^\circ\). (d) Slopes are negative reciprocals.

Step 1: Show that the slope is the tangent of the angle

For a line, the slope \(m\) is defined as the rise over run, or the change in \(y\) over the change in \(x\). The angle \(\theta\) is formed between the line and the \(x\)-axis. In a right-angled triangle formed by the line and the axes, the tangent of \(\theta\) (\(\tan \theta\)) is the ratio of the side opposite to \(\theta\) and the adjacent side (run), which is the same as the definition of slope: rise/run. Thus, \(m = \tan \theta\).

Step 2: Express angle between two lines using slopes

The angle \(\psi\) between two lines with slopes \(m_1\) and \(m_2\) can be found using the formula \(\tan \psi = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|\). This formula is derived from the tangent addition formula, considering the tangent of the angle between the lines is related to the difference of their slopes divided by one plus the product of their slopes.

Step 3: Calculate the acute angle between two specific lines

Given lines \(y = \frac{1}{3}x + 1\) and \(y = -\frac{1}{2}x - 3\), the slopes \(m_1 = \frac{1}{3}\) and \(m_2 = -\frac{1}{2}\) can be identified. Plug these into the formula for \(\tan \psi\): \(\tan \psi = \left| \frac{-\frac{1}{2} - \frac{1}{3}}{1 + \frac{1}{3} \cdot (-\frac{1}{2})} \right|\). Simplifying gives \(\tan \psi = \frac{5}{11}\). Then compute \(\psi\) using inverse tangent to find \(\psi \approx 24.44^\circ\).

Step 4: Show that perpendicular lines have negative reciprocal slopes

If two lines are perpendicular, then the angle \(\psi\) between them is \(90^\circ\), for which \(\tan \psi = \infty\). Using \(\tan \psi = \frac{m_2 - m_1}{1 + m_1 m_2}\), the denominator must be zero for \(\tan \psi\) to approach infinity. This implies \(1 + m_1 m_2 = 0\), showing \(m_2 = -\frac{1}{m_1}\), confirming they are negative reciprocals.

Key Concepts

Angle of Elevation

In trigonometry, the angle of elevation is the angle between the horizontal and the line of sight when looking upward at an object. Imagine standing at ground level and looking up at the top of a building. The angle your line of sight makes with the imaginary flat ground is the angle of elevation.
This concept is commonly used in problems where you need to determine the height of an object using the distance from the object and the angle of elevation.

• The angle of elevation is always measured from the horizontal.
• It plays a key role when calculating heights and distances using trigonometric ratios like tangent, sine, and cosine.

Overall, understanding the angle of elevation allows you to apply trigonometric functions effectively to solve real-world and mathematical problems.

Slope of a Line

The slope of a line is a measure of its steepness and is usually represented by the letter "m." It is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between any two points on the line. This is often expressed as "rise over run."
In mathematical terms, if you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line, the slope \( m \) is calculated as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

• A positive slope means the line rises as you move from left to right.
• A negative slope indicates the line falls as you move from left to right.
• A zero slope represents a horizontal line, and an undefined slope represents a vertical line.

Understanding the slope is crucial because it helps in analyzing and graphing linear equations, understanding rates of change, and interpreting the physical world via mathematical models.

Tangent of an Angle

The tangent of an angle in a right triangle is a fundamental concept in trigonometry. It relates the angle to the lengths of the sides of the triangle. Specifically, the tangent of an angle \( \theta \) is the ratio of the length of the side opposite the angle to the length of the side adjacent to it. In equation form, it is written as:\[ \tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} \]This trigonometric function is significant because it forms the basis of understanding slopes and angles in geometry and trigonometry.

• The tangent function is periodic with a period of 180 degrees or \( \pi \) radians.
• It is undefined at 90 degrees and multiples thereof, where the opposite side would be at a maximum while the adjacent is zero.
• For small angles, the tangent of an angle is approximately the same as the angle itself, when the angle is measured in radians.

This concept is essential in fields ranging from engineering to physics, where the relationship between angles and side lengths comes into play.

Perpendicular Lines

Two lines are said to be perpendicular when they intersect at a right angle, which is precisely 90 degrees. In terms of slopes, if two lines are perpendicular, the product of their slopes is -1. This is because one slope is the negative reciprocal of the other. Mathematically, if a line has a slope \( m_1 \), then a line perpendicular to it will have a slope \( m_2 \) such that:\[ m_1 \times m_2 = -1 \]

• If \( m_1 = a \), then \( m_2 = -\frac{1}{a} \).
• Horizontal lines, having zero slopes, are perpendicular to vertical lines, which have undefined slopes.

This property is vital in geometry for constructing perpendicular bisectors, proving the congruence of angles, and in design and layout tasks where right angles are required.