Problem 37 - Advanced Engineering Mathematics

Question

Use Cramer's rule to solve the system $$ \begin{aligned} &X=x \cos \theta+y \sin \theta \\ &Y=-x \sin \theta+y \cos \theta \end{aligned} $$ for \(x\) and \(y\).

Step-by-Step Solution

Answer

x = X\cos\theta - Y\sin\theta, y = Y\cos\theta + X\sin\theta.

Step 1: Identifying the Matrices

First, we identify the coefficients in the given equations. Rewrite the system:\[X = x\cos\theta + y\sin\theta\]\[Y = -x\sin\theta + y\cos\theta\]The coefficient matrix \( A \) is:\[ A = \begin{pmatrix} \cos\theta & \sin\theta \ -\sin\theta & \cos\theta \end{pmatrix} \] and the constants matrix is \( \begin{pmatrix} X \ Y \end{pmatrix} \). The variables matrix is \( \begin{pmatrix} x \ y \end{pmatrix} \).

Step 2: Calculate the Determinant of the Coefficient Matrix

To use Cramer's rule, compute the determinant of matrix \( A \):\[ \text{det}(A) = \cos\theta \cdot \cos\theta - (-\sin\theta) \cdot \sin\theta = \cos^2\theta + \sin^2\theta \]Using the Pythagorean identity, \( \cos^2\theta + \sin^2\theta = 1 \). So, \( \text{det}(A) = 1 \).

Step 3: Calculate Determinants for x and y

Calculate the determinants by replacing the columns of the coefficient matrix with the constants vector. For \( x \), replace the first column:\[ A_x = \begin{pmatrix} X & \sin\theta \ Y & \cos\theta \end{pmatrix} \]\[ \text{det}(A_x) = X\cos\theta - Y\sin\theta \]For \( y \), replace the second column:\[ A_y = \begin{pmatrix} \cos\theta & X \ -\sin\theta & Y \end{pmatrix} \]\[ \text{det}(A_y) = Y\cos\theta + X\sin\theta \].

Step 4: Use Cramer's Rule to Solve for x and y

Cramer's Rule gives:\[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = X\cos\theta - Y\sin\theta \]\[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = Y\cos\theta + X\sin\theta \].

Key Concepts

Matrix Determinants

A matrix determinant is a special number that can be calculated from a square matrix, and it plays a crucial role in solving systems of linear equations. It's like a DNA fingerprint for matrices, giving you vital information about whether a matrix is invertible, which means its inverse matrix exists.
For a 2x2 matrix like \( A \), the determinant is determined by the formula:

• \( \text{det}(A) = ad - bc \)
• where \( a, b, c, \text{ and } d \) are elements of the matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \).

Calculating the determinant gives you a single value. If that value is not zero, it means the matrix is invertible and solutions can be found for the linear system using methods like Cramer's Rule.
In our example exercise, you calculated the determinant of the coefficient matrix to be 1. This confirms the matrix is invertible, allowing us to solve for the variables \( x \) and \( y \).

Linear Systems

Linear systems are collections of linear equations involving the same set of variables. These equations can be solved to find values for these variables that satisfy all the equations in the system simultaneously. For example, the set of equations given in the original exercise represents a linear system.
Solving linear systems can involve various methods, including substitution, elimination, and matrix-related techniques such as Cramer's Rule.
Cramer's Rule is particularly useful because:

• It uses determinants to find the solution.
• It's applied to systems with the same number of equations as variables (i.e., square systems).

In our exercise, the linear system describes a rotation transformation, and Cramer's Rule helps find explicit solutions for \( x \) and \( y \) in terms of \( X \), \( Y \), and the trigonometric functions of \( \theta \).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. They are incredibly helpful for simplifying expressions and solving equations involving angles.
The Pythagorean identity is one fundamental trigonometric identity:

• \( \cos^2\theta + \sin^2\theta = 1 \)

This identity is essential for simplifying terms involving trigonometric functions, as in our exercise where it confirms that the determinant of the coefficient matrix equals 1.
In our problem, the trigonometric functions \( \cos \theta \) and \( \sin \theta \) represent the rotation of a vector in the plane by an angle \( \theta \). These functions help transform the original system's variables, \( x \) and \( y \), through rotation, expressed via the equations given.
Understanding these identities is key to simplifying complex expressions involving angles and making problems more manageable.