Problem 9 - Precalculus

Question

Find \(\sin \theta\) and \(\cos \theta,\) where \(\theta\) is the (acute) angle of rotation that eliminates the \(x^{\prime} y^{\prime}\) -term. Note: You are not asked to graph the equation. $$x^{2}-24 x y+8 y^{2}-8=0$$

Step-by-Step Solution

Answer

\( \sin \theta = \frac{3}{5} \), \( \cos \theta = \frac{4}{5} \).

Step 1: Identify coefficients of the quadratic form

The given equation is \( x^2 - 24xy + 8y^2 - 8 = 0 \). From this, identify the coefficients: \( A = 1 \), \( B = -24 \), and \( C = 8 \).

Step 2: Calculate the angle of rotation using tangent

The angle \( \theta \) to eliminate the \(xy\) term in the rotated coordinate system satisfies \( \tan 2\theta = \frac{B}{A-C} \). Substitute the values: \( \tan 2\theta = \frac{-24}{1-8} = \frac{-24}{-7} = \frac{24}{7} \).

Step 3: Find \( \sin 2\theta \) and \( \cos 2\theta \)

Since \( \tan 2\theta = \frac{24}{7} \), we use the identity \( \tan 2\theta = \frac{ \sin 2\theta }{ \cos 2\theta } \). Let \( \sin 2\theta = \frac{24}{\sqrt{24^2 + 7^2}} = \frac{24}{25} \) and \( \cos 2\theta = \frac{7}{\sqrt{24^2 + 7^2}} = \frac{7}{25} \).

Step 4: Calculate \( \sin \theta \) and \( \cos \theta \) using half-angle formulas

Using half-angle identities, we calculate:\[ \sin \theta = \sqrt{\frac{1 - \cos 2\theta}{2}} = \sqrt{\frac{1 - \frac{7}{25}}{2}} = \sqrt{\frac{18}{50}} = \frac{3}{5} \]\[ \cos \theta = \sqrt{\frac{1 + \cos 2\theta}{2}} = \sqrt{\frac{1 + \frac{7}{25}}{2}} = \sqrt{\frac{32}{50}} = \frac{4}{5} \]

Step 5: Verify the signs of \( \sin \theta \) and \( \cos \theta \)

Since \( \theta \) is an acute angle, both \( \sin \theta \) and \( \cos \theta \) are positive. The computed values \( \frac{3}{5} \) and \( \frac{4}{5} \) are both positive, consistent with an acute angle in the first quadrant.

Key Concepts

Quadratic Form

When dealing with equations of the form \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\]this represents a quadratic equation in two variables. The quadratic form is essential in analyzing conic sections like ellipses and hyperbolas. The form of the quadratic equation in this exercise is: \[x^2 - 24xy + 8y^2 - 8 = 0.\]From this, the coefficients are identified as:

• A = 1 (coefficient of \(x^2\)),
• B = -24 (coefficient of \(xy\)),
• C = 8 (coefficient of \(y^2\)).

The term \(xy\) is crucial in determining the orientation of the conic section, linked directly to the angle of rotation. This quadratic form is altered by eliminating the \(xy\)-term through a rotational method, aimed at simplifying the equation by aligning axes with its principal orientation.

Tangent Identity

To eliminate the \(xy\) term, it is necessary to calculate the angle of rotation \(\theta\). This is where the tangent identity comes into play: \[\tan 2\theta = \frac{B}{A-C}. \]By substituting the identified coefficients, \[\tan 2\theta = \frac{-24}{1 - 8} = \frac{24}{7}. \]This tangent identity helps us find the angle such that the rotated coordinate system will have no \(xy\)-term, simplifying the form of the equation. Calculating \(\tan 2\theta\) determines the direction and extent of rotation needed for the transformation of the coordinate system, setting the stage for subsequent computations of trigonometric functions.

Half-Angle Formulas

Once \(\tan 2\theta\) is known, the next step involves determining the half-angle trigonometric functions, \(\sin \theta\) and \(\cos \theta\), using the half-angle formulas. These formulas are: \[\sin \theta = \sqrt{\frac{1 - \cos 2\theta}{2}} \]\[\cos \theta = \sqrt{\frac{1 + \cos 2\theta}{2}}. \]From the given solution:

• \(\sin 2\theta = \frac{24}{25}\)
• \(\cos 2\theta = \frac{7}{25}\)

Substituting these into the half-angle formulas produces:

• \(\sin \theta = \frac{3}{5}\)
• \(\cos \theta = \frac{4}{5}\)

Since \(\theta\) is an acute angle, both values are positive, confirming the correctness of these calculations. The half-angle formulas are crucial for breaking down and simplifying the double angle metrics.

Coordinate System

The concept of rotation in a coordinate system is crucial for eliminating the \(xy\)-term, simplifying equations of conic sections. By rotating the coordinate axes, the quadratic equation is transformed into a simpler form. This method involves finding and applying the correct angle of rotation \(\theta\), thus aligning the axes with the principal orientations of the conic sections.In this exercise, the original coordinate axes are rotated to eliminate the cross-product term \(xy\). This forms a new coordinate system in terms of \(x'\) and \(y'\), where the equation becomes dependent on the new variables, making it easier to analyze and understand the geometry and properties of the shape.Hence, mastering rotation in coordinate systems not only optimizes solving quadratic equations but also enhances your ability to visualize problems geometrically.