Problem 35 - Algebra and Trigonometry

Question

Solve the equations $$ \begin{aligned} x &=X \cos \phi-Y \sin \phi \\ y &=X \sin \phi+Y \cos \phi \end{aligned} $$ 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: Multiply the Equations

Start by multiplying the first equation \( x = X \cos \phi - Y \sin \phi \) by \( \cos \phi \) and the second equation \( y = X \sin \phi + Y \cos \phi \) by \( \sin \phi \). This yields: \( x \cos \phi = X \cos^2 \phi - Y \sin \phi \cos \phi \) and \( y \sin \phi = X \sin^2 \phi + Y \sin \phi \cos \phi \).

Step 2: Add the Equations

Add the two results from Step 1: \( x \cos \phi + y \sin \phi = X(\cos^2 \phi + \sin^2 \phi) \). The terms involving \(Y\) cancel out due to opposing signs. Simplifying gives: \( x \cos \phi + y \sin \phi = X \). Use the Pythagorean identity \( \cos^2 \phi + \sin^2 \phi = 1 \). Thus, you get \( X = x \cos \phi + y \sin \phi \).

Step 3: Solve for Y

To solve for \(Y\), multiply the first equation by \( -\sin \phi \) and the second by \( \cos \phi \): \( -x \sin \phi = -X \cos \phi \sin \phi + Y \sin^2 \phi \) and \( y \cos \phi = X \cos \phi \sin \phi + Y \cos^2 \phi \).

Step 4: Subtract the Equations

Subtract the first transformed equation from the second: \( y \cos \phi - (-x \sin \phi) = Y(\cos^2 \phi + \sin^2 \phi) \). Simplifying, you find \( y \cos \phi + x \sin \phi = Y \). Thus, \( Y = y \cos \phi - x \sin \phi \).

Step 5: Results for X and Y

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

Key Concepts

System of Equations

When tackling a problem involving a system of equations, think about having multiple mathematical expressions that share common variables. These equations describe relationships between these variables. Our goal is to find values for these variables that satisfy all of the given equations. In this exercise, we have two trigonometric equations involving the variables \( x \), \( y \), \( X \), and \( Y \). The aim is to express \( X \) and \( Y \) in terms of \( x \) and \( y \).

To achieve this, we must cleverly manipulate the equations. This often involves steps like adding or subtracting the equations, multiplying them by strategic factors, or substituting values. In the solution provided, we used trigonometric identities and algebraic manipulation to eliminate \( Y \) initially to solve for \( X \). Then, a similar strategy helped solve for \( Y \) by eliminating \( X \).

Approaching systems of equations requires patience and practice. Try to look for symmetries or common terms which can be simplified or eliminated. This will make finding a solution less daunting and more systematic. As you practice more, these processes will become second nature.

Trigonometric Identities

Trigonometric identities are relationships involving trigonometric functions that hold true for all values of the involved variables. They are incredibly useful in simplifying expressions and solving equations involving trigonometric functions.

In our exercise, the Pythagorean identity \( \cos^2 \phi + \sin^2 \phi = 1 \) is the cornerstone for this problem. It helps simplify the equation by reducing terms and finding the expressions for \( X \) and \( Y \). This identity ensures the terms without \( Y \) in the calculations cancel out, making the equation easier to solve.
Understanding and memorizing these identities allow you to manipulate equations more effectively by reducing complex expressions to simpler forms. They serve as powerful tools in not only solving equations but also deriving formulas and proving other identities.

Coordinate Transformation

Coordinate transformation involves changing the coordinates of a point within one coordinate system to another. This transformation often uses trigonometric functions when dealing with angles and rotations.

In the given equations, the transformation changes coordinates from \( (X, Y) \) to \( (x, y) \) by rotating them through an angle \( \phi \). This reflects how the eternal mathematical concepts underlie physical transformations like rotation and translation. The equations given represent a rotation transformation, where \( X \) and \( Y \) relate to new rotated coordinates \( x \) and \( y \).
Understanding these transformations is essential in physics, computer graphics, and many engineering fields. They allow us to model and manipulate physical phenomena, simulate scenarios, and analyze systems from different perspectives. Through coordinate transformations, we gain the flexibility to choose coordinate systems that simplify our analysis of complex problems.