Problem 50 - 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 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 $$ y=\frac{1}{3} x+1 \quad \text { and } \quad y=-\frac{1}{2} x-3 $$ (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) The slope is \( m = \tan(\theta) \). (b) \( \tan(\psi) = \frac{m_2 - m_1}{1+ m_1 m_2} \). (c) The angle is \( 45^\circ \). (d) Perpendicular slopes are negative reciprocals.

Step 1: Understanding the Slope as Tangent of Angle

The slope of a line in the plane is defined as the tangent of the angle \( \theta \) formed between the line and the positive direction of the x-axis. Specifically, if a line makes an angle \( \theta \) with the x-axis, then the slope \( m \) of the line is given by \( m = \tan(\theta) \). This can be observed from the fact that in a right triangle, the tangent of an angle is the ratio of the opposite side over the adjacent side.

Step 2: Deriving the Formula for Angle Between Two Lines

Given two lines with slopes \( m_1 \) and \( m_2 \), the tangent of the angle \( \psi \) between them can be derived using the formula: \( \tan(\psi) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \). This formula is derived from trigonometric identities where the angle \( \psi \) is the difference between the angles of the two lines.

Step 3: Applying Formula to Specific Lines

For the lines \( y=\frac{1}{3}x + 1 \) and \( y=-\frac{1}{2}x - 3 \), first determine their slopes: \( m_1 = \frac{1}{3} \) and \( m_2 = -\frac{1}{2} \). Using the earlier formula: \( \tan(\psi) = \left| \frac{-\frac{1}{2} - \frac{1}{3}}{1 + (\frac{1}{3})(-\frac{1}{2})} \right| = \left| \frac{-\frac{3}{6} - \frac{2}{6}}{1 - \frac{1}{6}} \right| = \left| \frac{-\frac{5}{6}}{\frac{5}{6}} \right| = 1 \). Thus, \( \psi = \tan^{-1}(1) = 45^\circ \).

Step 4: Proving Perpendicular Slopes Are Negative Reciprocals

For lines to be perpendicular, the angle between them, \( \psi \), must be 90 degrees. From the tangent formula, if \( \tan(\psi) = \frac{m_2 - m_1}{1 + m_1 m_2} \) and \( \psi = 90^\circ \), then \( \tan(\psi) \to \infty \), indicating \( 1 + m_1 m_2 = 0 \). Therefore, it follows that \( m_1 m_2 = -1 \), demonstrating that the slopes are negative reciprocals of each other.

Key Concepts

Angle between two lines

The angle between two lines is not as simple as it might appear at first glance. This angle, usually called \( \psi \), is determined when two non-parallel lines intersect. To find this angle accurately, especially when dealing with slopes of lines, we use trigonometry. The tangent of the angle \( \psi \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by the equation:

• \( \tan(\psi) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \)

This formula arises from trigonometric identities, specifically focusing on the tangent of the difference between their angles.
It is important to recognize that this formula is derived assuming that the lines are in a plane and neither line is vertical, as the tangent would be undefined. The absolute value ensures that you always get the acute angle between the lines, making the solution universally applicable to any two intersecting, non-parallel lines.

Tangent of an angle

The tangent of an angle is a fundamental concept in trigonometry, especially in relation to lines. In a right triangle, the tangent of an angle \( \theta \) is defined as the ratio of the opposite side to the adjacent side.
Thus, if you are working with a line that makes an angle \( \theta \) with the x-axis, the slope \( m \) of that line can be expressed as \( \tan(\theta) \). This means:

• \( m = \tan(\theta) \)

Understanding this relationship is crucial for analyzing angles and slopes in a plane. It links algebra to geometry, allowing problems involving the angle to be approached with trigonometry. This concept is the backbone for deriving more complex results, such as the angle between two lines.
Remember, the tangent function is infinite (undefined) whenever the angle is \( 90^\circ \) or \( 270^\circ \), correlating with vertical lines.

Perpendicular lines

Perpendicular lines intersect at a right angle, or \( 90^\circ \). This special relationship between slopes can be examined using the tangent formula.
For two lines to be perpendicular, their slopes must satisfy the condition that \( m_1 m_2 = -1 \). This condition is effectively saying that the product of their slopes is negative one, which directly follows from:

• The slopes being negative reciprocals of each other.

When two lines are perpendicular, the angle \( \psi \) between them is \( 90^\circ \), and \( \tan(\psi) \) becomes undefined or approaches infinity. That’s because the tangent of \( 90^\circ \) tends towards infinity, revealing the negative reciprocal relationship.
Understanding this helps in recognizing perpendicularity without directly measuring angles or using advanced geometry tools. It provides a quick algebraic method to check if two lines are indeed perpendicular based solely on their slopes.

Trigonometric identities

Trigonometric identities play a pivotal role in simplifying and solving problems involving angles and lines. In the context of slope and angle between two lines, these identities become invaluable. One of the key identities used here includes:

• The tangent of the difference between two angles: \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \).

This identity helps derive the formula for the angle between two lines based on their slopes. Additionally, trigonometric identities such as the reciprocal identities are useful when dealing with perpendicular lines and their properties.
Having a firm grasp on these identities allows for a deeper understanding of problems involving angles formed by lines, whether on paper or interpreted through graphs. They bridge the gap between conceptual geometry and practical algebra, making problem-solving in these areas much more manageable.