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Chapter 9: Problem 34

Determine the angle of rotation necessary to transform the equation in \(x\) and \(y\) into an equation in \(X\) and \(Y\) with no \(X Y\) -term. $$4 x^{2}+2 x y+2 y^{2}-7=0$$

Short Answer

Expert verified
The angle of rotation is \(\theta = \frac{\pi}{8}\).

Step by step solution

01

Identify Coefficients for the Rotation Formula

First, identify the coefficients needed for the rotation formula. From the equation \(4x^2 + 2xy + 2y^2 - 7 = 0\), determine that \(A = 4\), \(B = 2\), and \(C = 2\). These coefficients are used to find the angle \( \theta \) needed to eliminate the \(XY\) term after rotation.
02

Use Rotation Angle Formula

The angle of rotation \(\theta\) is found using the formula \(\cot(2\theta) = \frac{A-C}{B}\). Substitute \(A = 4\), \(B = 2\), and \(C = 2\) into the formula to get \(\cot(2\theta) = \frac{4-2}{2} = 1\).
03

Solve for \(2\theta\)

To find \(2\theta\), set the equation \(\cot(2\theta) = 1\). Since \(\cot(2\theta) = 1\), \(2\theta\) corresponds to \(\frac{\pi}{4}\). This implies \(2\theta = \frac{\pi}{4}\).
04

Calculate \(\theta\)

To find \(\theta\), divide \(2\theta = \frac{\pi}{4}\) by 2. Thus, \(\theta = \frac{\pi}{8}\). This is the angle of rotation required.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotation Formula
When dealing with transformations in mathematics, particularly involving conic sections, the rotation formula plays an essential role. It helps in eliminating the cross-product term in equations for clearer interpretation. The general idea is to find an angle \(\theta\) such that the \(XY\) term of a transformed equation can be removed.
First, identify the coefficients \(A\), \(B\), and \(C\) from the original equation. These correspond to the coefficients of \(x^2\), \(xy\), and \(y^2\) respectively. For the problem given, these are \(A = 4\), \(B = 2\), and \(C = 2\).
The angle \(\theta\) for rotation can be calculated using the formula:
  • \(\cot(2\theta) = \frac{A-C}{B}\)
By solving this equation, you'll determine the required rotation angle that simplifies the conic section by eliminating the \(XY\) term.
Conic Sections
Conic sections refer to the curves obtained by intersecting a plane with a cone. These curves are quite common in geometry and include ellipses, parabolas, hyperbolas, and circles. When equations describing these curves include a cross-term like \(XY\), it can indicate that the axes of the graph are rotated.
For instance, in the equation given, \(4x^2 + 2xy + 2y^2 - 7 = 0\), the presence of the \(2xy\) term is a signal that rotation might be necessary to make the analysis easier. By rotating the axes, one can sometimes transform such an equation into a more standard form without the cross-product term, thus simplifying both the understanding and solving process.
Feel free to explore more into each specific conic section; each type of conic has its unique properties and applications in both theoretical and applied mathematics.
Trigonometry
Trigonometry is profoundly useful in mathematics, especially for transformations involving rotations. When rotating axes to transform equations, understanding trigonometric functions like sine, cosine, or cotangent becomes important. Remember, cotangent is the reciprocal of the tangent function, and it's useful when calculating angles in right triangles or rotations.
In the rotation formula, the use of \(\cot(2\theta)\) is crucial. To solve for \(2\theta\), use the inverse trigonometric functions or understand the basic trigonometric ratios. For example, knowing that \(\cot(\frac{\pi}{4}) = 1\) can guide you to quickly solve such equalities.
Practicing trigonometric identities and their relationships will enhance your problem-solving techniques, providing a solid foundation for tackling rotation problems in geometry and beyond.

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Most popular questions from this chapter

Consider the parametric curve \(x=a \ln t, y=\ln (b t), t>0\) Assume that \(b\) is a positive integer and \(a\) is a positive real number. Determine the Cartesian equation.

The product of two numbers is equal to the reciprocal of the difference of their reciprocals. The product of the two numbers is \(72 .\) Find the numbers.

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