Problem 35 - Precalculus: Mathematics for Calculus

Question

Rotation of Axes Formulas Solve the equations $$\begin{array}{l}x=X \cos \phi-Y \sin \phi \\\y=X \sin \phi+Y \cos \phi\end{array}$$ for \(X\) and \(Y\) in terms of \(x\) and \(y .\) [Hint: To begin, multiply the first equation by \(\cos \phi\) and the second by \(\sin \phi,\) and then add the two equations to solve for \(X .]\)

Step-by-Step Solution

Answer

\( X = x \cos \phi + y \sin \phi \), \( Y = y \cos \phi - x \sin \phi \).

Step 1: Expand Transformation

The given transformation equations are: 1. \( x = X \cos \phi - Y \sin \phi \)2. \( y = X \sin \phi + Y \cos \phi \). Our goal is to express \( X \) and \( Y \) in terms of \( x \) and \( y \).

Step 2: Multiply Equations

Multiply the first equation by \( \cos \phi \):\[ x \cos \phi = X \cos^2 \phi - Y \sin \phi \cos \phi \].Multiply the second equation by \( \sin \phi \):\[ y \sin \phi = X \sin^2 \phi + Y \sin \phi \cos \phi \].

Step 3: Add Equations

Add the two equations obtained from Step 2:\[ x \cos \phi + y \sin \phi = X \cos^2 \phi + X \sin^2 \phi \].Since \( \cos^2 \phi + \sin^2 \phi = 1 \), it simplifies to:\[ x \cos \phi + y \sin \phi = X \].

Step 4: Solve for X

From the simplified equation, we directly obtain:\[ X = x \cos \phi + y \sin \phi \].

Step 5: Use Similar Strategy for Y

Multiply the first original equation by \( \sin \phi \) and the second by \( \cos \phi \):1. \( x \sin \phi = X \cos \phi \sin \phi - Y \sin^2 \phi \)2. \( y \cos \phi = X \sin \phi \cos \phi + Y \cos^2 \phi \).

Step 6: Subtraction Leads to Y

Subtract the first newly derived equation from the second:\[ y \cos \phi - x \sin \phi = Y (\cos^2 \phi + \sin^2 \phi) \].Since \( \cos^2 \phi + \sin^2 \phi = 1 \), it simplifies to:\[ Y = y \cos \phi - x \sin \phi \].

Step 7: Final Expressions

The expressions for \( X \) and \( Y \) in terms of \( x \) and \( y \) are:- \( X = x \cos \phi + y \sin \phi \)- \( Y = y \cos \phi - x \sin \phi \)

Key Concepts

Coordinate Transformation

Coordinate transformation is a powerful mathematical tool used in various disciplines. It allows us to switch between different coordinate systems. Imagine you are looking at a point in a 2D plane. The position of this point can be described using different systems of coordinates, such as Cartesian (with coordinates \(x\) and \(y\)) or in a rotated system (often represented as \(X\) and \(Y\)).

In the context of this exercise, we focus on the rotation of axes. This involves the transformation equations:

• \( x = X \cos \phi - Y \sin \phi \)
• \( y = X \sin \phi + Y \cos \phi \)

These equations describe how a point changes position when the coordinate axes themselves are rotated by an angle \(\phi\). Our task is to find \(X\) and \(Y\) in terms of \(x\) and \(y\). This means reversing the transformation to express the original or new rotated coordinates in terms of the known initial or rotated ones.

Applications of coordinate transformations are widespread, including engineering, physics, and computer graphics, where they help transition smoothly between different frames of reference.

Trigonometric Identities

Trigonometric identities are foundational aspects of trigonometry. They help simplify expressions and solve equations that involve trigonometric functions.

A particularly important identity is the Pythagorean identity given by:

• \( \cos^2 \phi + \sin^2 \phi = 1 \)

This identity is crucial when dealing with rotations, as seen in the exercise solution.

When we multiply and add equations involving cosines and sines, this identity simplifies the transformations. For instance, in the step where we solve for \(X\), the equation \( x \cos \phi + y \sin \phi \) reduces to \( X \) because of this identity.

Understanding and using these identities not only simplifies calculations but also reveals insightful geometric interpretations like the concept of unit circles and the nature of the rotation angle \(\phi\).

This makes trigonometric identities invaluable for tackling a wide range of math problems, especially those involving angles and periodic phenomena.

Precalculus Exercises

Precalculus serves as a bridge between algebra and calculus. Exercises often involve various functions, transformations, and identities. The rotation of axes problem is a classic precalculus topic.

In such exercises, students engage with algebraic manipulations and trigonometric functions, preparing them for more advanced calculus concepts.

The main goal is to familiarize students with the skills needed to handle complex equations confidently and to comprehend the geometric transformations involved.

• This involves understanding how transformations alter the graph of a function.
• Exploring how to convert equations from one form to another, typically involving parameter changes like rotation or translation.

These concepts not only reinforce algebraic proficiency but also help develop spatial reasoning and an intuitive understanding of mathematical principles used in higher mathematics and various fields of science and engineering.

In summary, precalculus exercises like the rotation of axes foster analytical thinking and problem-solving abilities necessary for future math courses.