Problem 15 - Algebra and Trigonometry

Question

Write the appropriate rotation formulas so that in a rotated system the equation has no \(x^{\prime} y^{\prime}\) -term. $$x^{2}+4 x y-2 y^{2}-1=0$$

Step-by-Step Solution

Answer

The rotation angle needed to eliminate the cross-product term is \(\theta = \frac{\pi}{2}\).

Step 1: Identify rotation of axes formulas

The rotation of axes formulas are \(x = x^{\prime} \cos(\theta) - y^{\prime} \sin(\theta)\) and \(y = x^{\prime} \sin(\theta) + y^{\prime} \cos(\theta)\) where \(x^{\prime}\) and \(y^{\prime}\) are the coordinates of the rotated system and \(\theta\) is the angle of rotation.

Step 2: Substitute the rotation formulas into the original equation

By substituting \(x\) and \(y\) with the formulas from step 1 in the given equation, it becomes \((x^{\prime} \cos(\theta) - y^{\prime} \sin(\theta))^2 +4(x^{\prime} \cos(\theta) - y^{\prime} \sin(\theta))(x^{\prime} \sin(\theta) + y^{\prime} \cos(\theta)) -2(x^{\prime} \sin(\theta) + y^{\prime} \cos(\theta))^2 -1 = 0\)

Step 3: Simplify and look for conditions

After simplifying, the \(x^{\prime} y^{\prime}\)-term will still be present but its coefficient can be influenced by choosing the right \(\theta\). To eliminate the term completely, the criterion is that the coefficient of \(x^{\prime} y^{\prime}\) has to be zero. This leads to the equation \(-4\cos(\theta)\sin(\theta) -4\cos(\theta)\sin(\theta) = 0\).

Step 4: Solve the equation for \(\theta\)

The derived equation in step 3 simplifies to \(-8\cos(\theta)\sin(\theta)=0\). This equation is true for \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\), etc. But since a rotation of \(\pi\) or \(\frac{2\pi}{3}\) will bring back the original axes, the smallest positive \(\theta\) for which \(8\cos(\theta)\sin(\theta) = 0\) is \(\theta = \frac{\pi}{2}\).

Key Concepts

Rotation of Axes

When dealing with certain geometric problems or equations, the rotation of axes can clarify and simplify our results. This technique changes the coordinate system such as the orientation of the axes, while preserving the relationship between points on the plane. It's like tilting your head to see a drawing at a different angle.

The formulas for rotating the axes involve the angle of rotation, \(\theta\), and are as follows:

• \(x = x^{\prime} \cos(\theta) - y^{\prime} \sin(\theta)\)
• \(y = x^{\prime} \sin(\theta) + y^{\prime} \cos(\theta)\)

Here, \((x^{\prime}, y^{\prime})\) represents the new coordinates after rotation. The aim is often to transform an equation so that certain terms, like the \(x^{\prime}y^{\prime}\)-term, disappear, simplifying the equation's interpretation.

Trigonometric Substitution

Trigonometric substitution comes into play after setting up the rotation formulas. It's a key strategy for transforming equations in mathematics. By substituting the expressions provided by the rotation of axes into the original equation, we involve trigonometric functions like sine and cosine.

This process can initially complicate the equation but is essential to further manipulate and ultimately simplify complex expressions. In our example, each instance of \(x\) and \(y\) in the original equation is replaced by its corresponding trigonometric expression. This substitution allows us to express every part of our equation in terms of \(\theta\) and the new axes, \(x^{\prime}\) and \(y^{\prime}\).

Equation Simplification

After substituting the rotation formulas into the given equation, the goal is to simplify it to make analysis easier. Simplifying means combining like terms and ensuring that the equation is as compact and readable as possible.

In this context, simplifying allows us to focus on the specific terms we want to manipulate, like eliminating the \(x^{\prime}y^{\prime}\)-term. The primary aim here is to set the coefficient of this term to zero. Through careful algebra and an understanding of trigonometric identities, we derive a new condition related to \(\theta\) that must be satisfied to remove this cross-term.

Solving for Angle Theta

With an equation representing the condition for term elimination, solving for \(\theta\) involves understanding trigonometric identities and roots. In this case, the equation was simplified to \(-8\cos(\theta)\sin(\theta) = 0\).

This leads directly to identifying values of \(\theta\) that make the expression zero, such as \(\theta = 0, \frac{\pi}{2}, \pi,\) etc. However, not all solutions are practically useful. Ideally, we opt for the smallest positive \(\theta\) that achieves our goal because it minimizes the rotation needed to eliminate the \(x^{\prime}y^{\prime}\)-term.

At the conclusion of this problem, the smallest useful \(\theta\) is determined to be \(\frac{\pi}{2}\), leading to an elegant geometric understanding without re-complicating the system with the original axis orientation.