Problem 7 - Algebra and Trigonometry

Question

Write the appropriate rotation formulas so that in a rotated system the equation has no \(x^{\prime} y^{\prime}\) -term. $$x^{2}+x y+y^{2}-10=$$

Step-by-Step Solution

Answer

The short answer is the calculated \( \theta \) that results in the elimination of the cross-term, and the final equation expressed in the rotated coordinates.

Step 1: Define Original Variables in terms of Rotated Variables

The primary step is to choose an angle of rotation \( \theta \). The original variables can be represented in terms of the new (rotated) variables using the simple rotation formulas: \( x = x'\cos(\theta) - y'\sin(\theta) \) and \( y = x'\sin(\theta) + y'\cos(\theta) \)

Step 2: Substitute and Expand

Substitute these into the given equation \( x^{2} + xy + y^{2} -10 = 0 \) and then expand.

Step 3: Find Angle Of Rotation

Consider the resulting equation as a second degree equation \( Ax'^2 + Bx'y' + Cy'^2 - 10 = 0 \). We want to eliminate the cross-term \( Bx'y' \). The cross term gets eliminated if we take \( tan(2\theta) = B/(A-C) \). From here, calculate the value of \( \theta \)

Step 4: Express In Rotated Coordinates

Now substitute back the calculated \( \theta \) to the original equation to express the equation in rotated coordinates.

Key Concepts

Coordinate System Rotation

The concept of coordinate system rotation is crucial in simplifying equations by changing the perspective from which we view them. Imagine you have a grid with an x and y-axis. Normally, these axes are fixed, but by rotating them, we can often make complex equations simpler. Rotation doesn't alter the actual shape or points of a graph; it merely changes the coordinates where you would find these points.

For instance, in our exercise, we have an equation that includes a term with both x and y. Rotating the coordinate system might allow us to view this equation from a new angle where the interaction between x and y is eliminated, making the equation easier to solve. It's a bit like tilting your head to see a picture from a different angle, and suddenly noticing new patterns that weren't obvious before.

Rotation Formulas

Rotation formulas provide the mathematical means to translate the original coordinates into new coordinates based on the rotation of axes. To rotate the coordinate system, we use the rotation angle \( \theta \) and apply trigonometric functions. The formulas are:\
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• \( x = x'\cos(\theta) - y'\sin(\theta) \)\
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• \( y = x'\sin(\theta) + y'\cos(\theta) \)\
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Here, \( (x, y) \) are the original coordinates and \( (x', y') \) are the coordinates after rotation. The angle \( \theta \) is actually what controls how far around we're rotating our system. By carefully choosing \( \theta \) we can manipulate the terms of the equation, particularly the 'cross term.' This process is central to understanding the impact of rotation in converting equations to a more manageable form. Remember, \( \cos(\theta) \) and \( \sin(\theta) \) are simply ratios, so they're taking parts of the original coordinates to create the new ones.

Elimination of Cross Term

The elimination of the cross term is often the objective when we decide to rotate the coordinate system. The 'cross term' in an equation is the part that multiplies x and y together—like the \( xy \) in our original equation. These can be tricky because they represent an interaction between the x and y variables that isn't as straightforward to solve as the terms involving just \( x^2 \) or \( y^2 \).

When we rotate the axes, we aim at making this cross term disappear. This is achieved by setting \( \tan(2\theta) = \frac{B}{A - C} \) where B is the coefficient of the cross term, and A and C are the coefficients of the \( x'^2 \) and \( y'^2 \) terms, respectively, in the new equation. By solving for \( \theta \) here, we find the precise angle that makes the cross term vanish. Once that's done, the equation simplifies to one that is much easier to handle, typically a standard form that can showcase important features like the orientation and shape of a conic section without the added complexity of the cross term.