Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, determine the angle \(\theta\) that will eliminate the \(x y\) term and write the corresponding equation without the \(x y\) term. \(x^{2}+3 \sqrt{3} x y+4 y^{2}+y-2=0\)

Step-by-Step Solution

Answer

The angle \( \theta \) is \( 60^{\circ} \), simplifying the equation removes the \( xy \) term.

Step 1: Identify the necessary transformation

The angle \( \theta \) that eliminates the \( xy \) term in a conic section equation can be found using the formula \( \cot 2\theta = \frac{A-C}{B} \). In this equation, \( A = 1 \), \( B = 3\sqrt{3} \), and \( C = 4 \).

Step 2: Apply the transformation formula

Substitute the values into the formula: \( \cot 2\theta = \frac{1 - 4}{3\sqrt{3}} = \frac{-3}{3\sqrt{3}} = -\frac{1}{\sqrt{3}} \).

Step 3: Solve for \( \theta \)

From \( \cot 2\theta = -\frac{1}{\sqrt{3}} \), recognize that when \( \cot 2\theta = -\frac{1}{\sqrt{3}} \), \( 2\theta = 120^{\circ} \). Thus, \( \theta = 60^{\circ} \).

Step 4: Use trigonometric identities

Using \( \theta = 60^{\circ} \), apply the rotation transformation formulas: \( x = x' \cos\theta - y' \sin\theta \) and \( y = x' \sin\theta + y' \cos\theta \) with \( \cos 60^{\circ} = \frac{1}{2} \) and \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \).

Step 5: Substitute and simplify

Replace \( x \) and \( y \) in the original equation with expressions involving \( x' \) and \( y' \):\[x = x' \cdot \frac{1}{2} - y' \cdot \frac{\sqrt{3}}{2},\quad y = x' \cdot \frac{\sqrt{3}}{2} + y' \cdot \frac{1}{2}\]and simplify using these substitutions to eliminate the \( xy \) term.

Key Concepts

Rotation of Axes

In geometry, rotating the axes is a technique used to simplify equations, often removing the cross-product \(xy\) term in conic sections. This process helps in shifts between different orientations without changing the inherent geometry of the shape. Using rotation of axes isn't just for aesthetic cleanup; it's about precision and clarity in understanding geometric identities. The method revolves around transforming coordinates from \((x, y)\) to \((x', y')\) using the angles specified by a particular formula.

• Equation transformation: \(x = x' \cos\theta - y' \sin\theta\) and \(y = x' \sin\theta + y' \cos\theta\).
• Purpose: Eliminate certain terms (like \(xy\)) to simplify the equation.

Recognize that rotation does not alter dimensions but helps frame them in a form that's easier to comprehend, akin to adjusting a painting to the perfect angle for viewing.

Angle of Elimination

To eliminate the \(xy\) term in an equation such as a conic section, we determine a specific angle called the angle of elimination. This angle can efficiently re-model the equation into a more recognizable and simplified form.To find this angle, we employ a special formula:\[\cot 2\theta = \frac{A-C}{B}\]Applying the values from the equation, where \(A\), \(B\), and \(C\) are coefficients of \(x^2\), \(xy\), and \(y^2\) respectively, provides the angle \(\theta\). Here's a breakdown:

• Identify the coefficients: \(A = 1\), \(B = 3\sqrt{3}\), and \(C = 4\).
• Substitute into the formula: \(\cot 2\theta = \frac{-3}{3\sqrt{3}} = -\frac{1}{\sqrt{3}}\).
• Solve for \(2\theta\), then \(\theta\), yielding \(\theta = 60^{\circ}\).

Once \(\theta\) is known, the equation can be rotated, nullifying the \(xy\) term and achieving a more streamlined configuration.

Trigonometric Identities

Trigonometric identities are the mathematical tools that allow us to compute angles and functions in various contexts, such as when rotating axes. They are key in expressing one function in terms of another to ease transformation calculations.When transforming a conic section equation to eliminate the \(xy\) term using rotation:

• Utilize basic identities: \(\cos 60^{\circ} = \frac{1}{2}\) and \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\).
• These values replace the trigonometric functions in the transformation formulas \(x = x' \cos\theta - y' \sin\theta\) and \(y = x' \sin\theta + y' \cos\theta\).

Breaking down these identities into simple components clarifies their transformative power. Therein lies the beauty: these identities, often introduced early in trigonometry, have fundamental applications in reshaping complex expressions in geometry.