Problem 57 - Calculus Early Transcendentals: Pearson New International Edition

Question

The graph of \(x \cos \alpha+y \sin \alpha=d\) is a line. Show that the perpendicular distance from the origin to this line is \(|d|\) by making a rotation of axes through the angle \(\alpha\).

Step-by-Step Solution

Answer

The distance from the origin to the line is \(|d|\).

Step 1: Identify the Problem

We are asked to show that the perpendicular distance from the origin to the line given by the equation \(x \cos \alpha + y \sin \alpha = d\) is \(|d|\) by utilizing a rotation of axes.

Step 2: Understanding Rotation of Axes

We know that a rotation of axes by an angle \(\alpha\) involves using the transformations \(x = x' \cos \alpha - y' \sin \alpha\) and \(y = x' \sin \alpha + y' \cos \alpha\). This will help us analyze the line equation in a rotated coordinate system where it simplifies.

Step 3: Substitute Rotation into the Line Equation

Substitute \(x = x' \cos \alpha - y' \sin \alpha\) and \(y = x' \sin \alpha + y' \cos \alpha\) into the line equation \(x \cos \alpha + y \sin \alpha = d\). This yields:\[(x' \cos \alpha - y' \sin \alpha) \cos \alpha + (x' \sin \alpha + y' \cos \alpha) \sin \alpha = d\]

Step 4: Simplify the Transformed Equation

After expanding and simplifying the equation:\[ x' \cos^2 \alpha - y' \sin \alpha \cos \alpha + x' \sin^2 \alpha + y' \sin \alpha \cos \alpha = d \]Combine terms to simplify:\[ x'(\cos^2\alpha + \sin^2\alpha) = d \]Using the identity \(\cos^2 \alpha + \sin^2 \alpha = 1\), we have:\[ x' = d \]

Step 5: Analyze the Simplified Equation

The simplified equation \(x' = d\) is a vertical line in the \(x'y'\)-coordinate system. This shows the distance of this line from the origin (0,0) in the rotated system, which is simply the absolute value of \(d\).

Step 6: Conclude with the Distance Result

Thus, the perpendicular distance from the origin to the line \(x \cos \alpha + y \sin \alpha = d\) is \(|d|\). This is confirmed by the equation \(x' = d\) in the rotated system.

Key Concepts

Rotation of Axes

A rotation of axes is a powerful mathematical technique used to simplify problems involving coordinates. It involves turning the entire coordinate system by a specific angle, allowing us to look at equations from a new perspective. Initially, points in the plane are described using the coordinates \((x, y)\). After rotating the axes by an angle \(\alpha\),we switch to coordinates \((x', y')\) using the transformations:

• \(x = x' \cos \alpha - y' \sin \alpha\)
• \(y = x' \sin \alpha + y' \cos \alpha\)

This essentially means we're rotating the entire plane \(\alpha\) degrees. Understanding the rotation of axes helps us analyze geometric figures by aligning them with the coordinate axes,making it easier to interpret and solve equations. This innovative method allows for clearer insights into mathematical problems and sometimes provides simpler formulae for complex equations.
For example, in the original problem, rotating the axes turns the line \(x \cos \alpha + y \sin \alpha = d\) into the much simpler equation \(x' = d\) in the new coordinate system,making the calculation of distances more straightforward.

Trigonometric Identities

Trigonometric identities are key to manipulating and simplifying equations in mathematics, especially when dealing with rotations and coordinates.In the context of the rotation of axes, one central identity plays a crucial role: \(\cos^2 \alpha + \sin^2 \alpha = 1\). This is the fundamental identity and is essential for simplifying expressions.For instance, in our exercise, to simplify the equation \((x' \cos \alpha - y' \sin \alpha) \cos \alpha + (x' \sin \alpha + y' \cos \alpha) \sin \alpha = d\),using the identity \(\cos^2 \alpha + \sin^2 \alpha = 1\) leads directly to \(x' = d\).
This simplification reveals how crucial trigonometric identities are in solving mathematical problems by reducing complexity and focusing on what's vital.Additionally, other identities, like the double angle formulas or sum-to-product formulas, can also assist in problems involving more complex transformations or equations.

Coordinate Transformation

Coordinate transformation is a process of changing the reference frame from which we observe a set of coordinates. This allows us to simplify equations or interpret them differently.In this exercise, transforming coordinates involved a rotation, which is one common type of transformation.By applying a rotation of axes, the coordinates \((x, y)\) are transformed to \((x', y')\) using specific formulas:

• \(x = x' \cos \alpha - y' \sin \alpha\)
• \(y = x' \sin \alpha + y' \cos \alpha\)

This specific transformation simplifies the problem statement to a form where it's easier to interpret and solve—in this case, converting\(x \cos \alpha + y \sin \alpha = d\) to \(x' = d\).
This shows the new equation in the transformed coordinate system and illustrates that the distance from the origin to the line is simply the absolute value of \(d\).Thus, coordinate transformations not only simplify calculations but also offer a different perspective, unveiling new solutions to familiar problems.