Problem 35 - Precalculus Mathematics for Calculus

Question

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: Multiply and Add Equations

Start by multiplying the first equation \(x = X \cos \phi - Y \sin \phi\) by \(\cos \phi\), giving: \[ x \cos \phi = X \cos^2 \phi - Y \sin \phi \cos \phi. \] Next, multiply the second equation \(y = X \sin \phi + Y \cos \phi\) by \(\sin \phi\), resulting in: \[ y \sin \phi = X \sin^2 \phi + Y \sin \phi \cos \phi. \] Add these two results together:\[ x \cos \phi + y \sin \phi = X(\cos^2 \phi + \sin^2 \phi). \] Recognizing that \(\cos^2 \phi + \sin^2 \phi = 1\), it simplifies to:\[ x \cos \phi + y \sin \phi = X. \]

Step 2: Solve for X

From the simplified equation:\[ X = x \cos \phi + y \sin \phi. \]We have now expressed \(X\) in terms of \(x\), \(y\), and \(\phi\).

Step 3: Multiply and Subtract Equations

To find \(Y\), return to the original system: multiply the first equation by \(\sin \phi\):\[ x \sin \phi = X \cos \phi \sin \phi - Y \sin^2 \phi. \]Multiply the second equation by \(\cos \phi\):\[ y \cos \phi = X \sin \phi \cos \phi + Y \cos^2 \phi. \]Subtract the first equation result from the second:\[ y \cos \phi - x \sin \phi = Y (\cos^2 \phi + \sin^2 \phi). \]Simplify using \(\cos^2 \phi + \sin^2 \phi = 1\):\[ y \cos \phi - x \sin \phi = Y. \]

Step 4: Solve for Y

From the simplified equation:\[ Y = y \cos \phi - x \sin \phi. \]Thus, \(Y\) is expressed in terms of \(x\), \(y\), and \(\phi\).

Key Concepts

Trigonometric Identities

Trigonometric identities are foundational tools in mathematics, helping to simplify expressions and solve equations. In this exercise, the primary identity used is the Pythagorean identity: \( \cos^2 \phi + \sin^2 \phi = 1 \). This identity is crucial because it allows us to reduce complex expressions and solve for unknowns in equations that involve trigonometric functions.
To apply this identity in solving the given system of equations, notice how both equations are manipulated by multiplying with either \( \cos \phi \) or \( \sin \phi \). This strategic move uses the properties of cosine and sine, combined with the identity, to combine terms effectively.

Applying Trigonometric Identities

• The purpose of multiplying each equation by a trigonometric function is to set up conditions where we can use the identity to simplify terms.
• Recognizing patterns in equations that allow for substitution or simplification using identities is key to overcoming challenges in algebra and trigonometry.
• In this example, applying \( \cos^2 \phi + \sin^2 \phi = 1 \) helps isolate \( X \) and \( Y \), making it easier to express these variables in terms of known quantities \( x \) and \( y \).

System of Equations

A system of equations involves two or more equations that share the same set of variables. Solving such systems is a fundamental skill in algebra. In this problem, we're looking to find values for \( X \) and \( Y \) using the provided equations involving \( x \) and \( y \). This exercise is a typical example of solving linear equations with trigonometric elements.
To solve the system, the goal is to manipulate the equations to isolate and find expressions for one variable at a time. We achieved this by using trigonometric identities and algebraic manipulation to eliminate terms and simplify the system sequentially.

Techniques in Solving Systems of Equations

• Using addition or subtraction: Combine equations in a manner that simplifies them or eliminates one variable, as seen when adding or subtracting the transformed equations.
• Substitution: Introduce or substitute known identities, as applied with trigonometric identities here, to simplify the system.
• This process ensures all variables are expressed with respect to known quantities, providing a clear solution path.

Algebraic Manipulation

Algebraic manipulation is at the heart of solving any mathematical problem. It involves rearranging equations, substituting values, and applying mathematical operations to simplify or solve for unknowns. In this exercise, algebraic manipulation helped to transition from one form of the equation to another, making it easier to solve for \( X \) and \( Y \).
The problem demonstrates algebraic techniques like multiplication, addition, and subtraction applied to equations in a systematic way. By doing so, equations are transformed into a single variable format, which allows easy extraction of values.

Strategies for Effective Algebraic Manipulation

• Step-by-step transformation: Take complex equations one step at a time, simplifying using basic algebraic rules.
• Always look for opportunities to use identities or known formulas to reduce complexity.
• Ensure accuracy at each step: Double-check operations like multiplication and simplification to maintain equation integrity.
• Using these methods effectively can turn seemingly complicated problems into manageable sets of equations that reveal solutions intuitively.