Problem 28 - Thomas Calculus

Question

Find the sine and cosine of an angle in Quadrant II through which the coordinate axes can be rotated to eliminate the cross product term from the equation $$4 x^{2}-4 x y+y^{2}-8 \sqrt{5} x-16 \sqrt{5} y=0$$ Do not carry out the rotation.

Step-by-Step Solution

Answer

\( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}, \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)

Step 1: Identify the Angle for Transformation

We need to find an angle \( \theta \) such that the axes are rotated, eliminating the cross product term from the conic equation. The general form for the rotation of axes involves setting the cross product coefficient \( B = 0 \). In the equation \( 4x^2 - 4xy + y^2 - 8\sqrt{5}x - 16\sqrt{5}y = 0 \), the coefficient of the \( xy \) term is \( -4 \). To eliminate this term, we need \( 4\sin(2\theta) = -4 \).

Step 2: Solve the Condition for the Angle

Set up the equation \( 4\sin(2\theta) = -4 \) to find \( \sin(2\theta) \). Solving, we have: \[ \sin(2\theta) = -1 \] This indicates that \( 2\theta = 3\frac{\pi}{2} + 2k\pi \) for any integer \( k \). Thus, \( \theta = \frac{3\pi}{4} + k\pi \).

Step 3: Determine Sine and Cosine of the Angle

Since we are working in Quadrant II, we select \( \theta = \frac{3\pi}{4} \) to find \( \sin(\theta) \) and \( \cos(\theta) \). For this angle: \[ \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \] \[ \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \] This matches the expected signs for sine and cosine in Quadrant II.

Key Concepts

Rotation of Axes

When dealing with equations, especially those involving conic sections like ellipses and hyperbolas, rotating the coordinate axes can simplify the equation by eliminating cross-product terms. This process is very useful for bringing equations to a form where it's easier to identify the type of conic section.The idea behind the rotation of axes is to find an angle \( \theta \) to transform the original axes into new ones. The transformation often revolves around the formula:

• \( x = x'\cos\theta - y'\sin\theta \)
• \( y = x'\sin\theta + y'\cos\theta \)

By finding the right \( \theta \), the cross-product term (like \( xy \) in our equation) can be nullified. In our example equation, the \( xy \) term had a coefficient of \(-4\). The goal was to set this to zero. The mathematical condition to eliminate the \( xy \) term is achieved by using \( 2\theta \), where \( \sin(2\theta) = \frac{-B}{A - C} \). This concept is fundamental when analyzing and transforming conic sections in standard form.

Conic Sections

Conic sections are fundamental geometric shapes that result from the intersection of a plane with a cone. They include circles, ellipses, parabolas, and hyperbolas. Recognizing and transforming these sections is key to understanding their properties.Each conic section has a general quadratic form: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]The presence of the \( xy \) term indicates that the axes might not align with the natural symmetry of the conic. In simpler terms, this means the conic section could be rotated with respect to the axes.To convert this general form into a simplified version without the \( xy \) term, we use the rotation of the axes. By eliminating the \( Bxy \) term, the equation can often simplify to look like:

• Ellipse: \( \frac{x'^2}{a^2} + \frac{y'^2}{b^2} = 1 \)
• Hyperbola: \( \frac{x'^2}{a^2} - \frac{y'^2}{b^2} = 1 \)

These forms now enable us to easily identify and study the properties of the conic sections.

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions to one another, like sine, cosine, and tangent. These identities are pivotal when dealing with transformations such as the rotation of axes.In the given problem, to eliminate the cross-product term, we identified that \( \sin(2\theta) = -1 \). This identity is a cornerstone in rotating the axes, directly linking the transformation angle \( \theta \) with the values of sine and cosine.Some key trigonometric identities used in these transformations are:

• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)
• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)

In Quadrant II, where \( \theta = \frac{3\pi}{4} \), these identities are essential because they allow us to compute the exact values for \( \sin(\theta) \) and \( \cos(\theta) \): \[ \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \] \[ \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \] Understanding these identities make it easier to determine the orientation and properties of conic sections.