Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, find a new representation of the given equation after rotating through the given angle. \(2 x^{2}+8 x y-1=0, \theta=30^{\circ}\)

Step-by-Step Solution

Answer

The rotated equation, in terms of new coordinates \( x' \) and \( y' \), results from substituting the rotation formulas into the original equation and simplifying.

Step 1: Understanding the Rotation Formula

Rotating a conic section involves replacing the variables in the equation using the rotation formulas. The formulas for rotating coordinates are given by:\[x' = x \cos \theta - y \sin \theta\]\[y' = x \sin \theta + y \cos \theta\]where \( \theta = 30^{\circ} \). We will use these to find the expression in terms of \( x' \) and \( y' \).

Step 2: Convert Trigonometric Values

Calculate \( \cos \theta \) and \( \sin \theta \) for \( \theta = 30^{\circ} \). From trigonometric ratios, we know that:\[ \cos 30^{\circ} = \frac{\sqrt{3}}{2}, \quad \sin 30^{\circ} = \frac{1}{2} \]

Step 3: Substitute and Derive New Variables

Substitute the values of \( \cos \theta \) and \( \sin \theta \) into the rotation formulas to get:\[x' = \frac{\sqrt{3}}{2}x - \frac{1}{2}y\]\[y' = \frac{1}{2}x + \frac{\sqrt{3}}{2}y\]Now, solve these equations for \( x \) and \( y \) in terms of \( x' \) and \( y' \).

Step 4: Express x and y in terms of x' and y'

Rearrange the equations from Step 3 to find expressions for \( x \) and \( y \):\[x = \frac{\sqrt{3}}{2}x' + \frac{1}{2}y'\]\[y = -\frac{1}{2}x' + \frac{\sqrt{3}}{2}y'\]

Step 5: Substitute Back into Original Equation

Substitute the expressions for \( x \) and \( y \) from Step 4 into the original equation \( 2x^2 + 8xy - 1 = 0 \). This involves substituting and expanding:- Substitute in for \(x\) to get \(x = \frac{\sqrt{3}}{2}x' + \frac{1}{2}y'\), and for \(y\) as \(y = -\frac{1}{2}x' + \frac{\sqrt{3}}{2}y'\).- Expand and simplify the equation.

Step 6: Simplify the New Equation

Simplify the equation using algebraic methods. Collect like terms and combine them into a neat expression. This encompasses finding coefficients and using algebraic identities where possible to simplify the expressions further.

Step 7: Present the Final Equation

After simplifying in Step 6, present the rotated equation in terms of \( x' \) and \( y' \). Ensure the expression is simplified to its lowest form to represent the rotated conic.

Key Concepts

Coordinate Geometry

Coordinate geometry is the study of geometric figures through algebra. In this exercise, we appreciate its usefulness for answering questions related to the transformation of shapes. Here, when rotating a conic section, coordinate geometry helps us handle the shift of axes. This means we use coordinates and algebraic expressions to describe and explore these transformations.

When you rotate a shape, you're essentially moving it around a central point or axis. In 2D geometry, this involves changing the position of every point along its trajectory. To achieve this for an equation, we use the rotation formulas for changing coordinates, which balance both geometry and algebra.

• Set up original coordinates: Use this as the starting reference.
• Apply rotation formulas: Determine new coordinates post rotation.
• Utilize trigonometric functions: These help in calculating sine and cosine for rotating the axes.

The goal is to transform the original equations into a new form following the rotation.

Trigonometric Substitution

Trigonometric substitution is a method used in various mathematical procedures to simplify expressions that involve trigonometric functions. For rotation of conic sections, it is crucial as it allows replacement of terms involving coordinate axes by new variables, depending on sine and cosine of the rotation angle.

In our problem, we substitute the trigonometric functions using \( heta = 30^\circ \), which simplifies the expressions since specific angles have known sine and cosine values. This substitution involves: \[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2} \]

By substituting these values into the rotation formulas, we essentially re-orient the conical equation. This step is critical, as it translates geometric information into a useful form for algebraic manipulation, facilitating further simplification.

Algebraic Simplification

Algebraic simplification is the process of transforming an equation into a simpler or more manageable form. In dealing with the rotation of conic sections, it comes after substituting new variables derived from trigonometric relationships.

The aim is to make the equation easier to understand or solve by consolidating like terms and simplifying complex expressions. This involves:

• Expanding substituted terms: Using algebraic identities to break down expressions.
• Combining like terms: Gather and simplify coefficients of same variables.
• Ensuring correctness: Regularly check simplification steps to avoid errors.

For example, after substitution, we expand expressions and simplify terms involving \( x' \text{ and } y' \) until we get a clean, concise equation. This makes further algebraic operations much easier to manage.

Equation Transformation

Equation transformation is the process of changing an equation to a new form, often to either simplify the problem or to analyze different properties of the original equation. In this context, transformation refers to rewriting equations in terms of new variables after a coordinate shift.

Initially, we write coordinates in terms of their rotated counterparts and substitute back into the original equation. For this exercise, we substitute back expressions for \( x \text{ and } y \) in terms of \( x' \text{ and } y' \). Here's what happens next:

• Substitute: Place expressions of \( x \) and \( y \) into the original equation.
• Expand the equation: Fully open up the terms affected by substitution.
• Simplify: Reduce this to its simplest expression using algebraic techniques.

The final transformation provides an insightful new perspective of the equation, shifting perceptions from one temporal form of the conic section to its rotated version. Such transformations are highly valued in mathematical modeling and analysis.