Problem 8 - 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

\(\sin\theta\) and \(\cos\theta\) are calculated using \(\theta = \frac{1}{2} \tan^{-1}\left(-\frac{24}{7}\right)\).

Step 1: Identify the Coefficient for Rotation

Identify the coefficient of the cross term (i.e., the `xy` term) in the given quadratic equation. In this case, the coefficient is 24.

Step 2: Use the Rotation Formula

To eliminate the `xy` term in a conic equation of the form \(Ax^2 + Bxy + Cy^2 = 0\), we use the angle formula \( \tan 2\theta = \frac{B}{A-C} \). In our equation, \(A = 1\), \(B = 24\), and \(C = 8\). Compute \( \tan 2\theta = \frac{24}{1-8} = -\frac{24}{7} \).

Step 3: Find 2θ Using Inverse Tangent

Now solve for \(2\theta\) using the inverse tangent: \(2\theta = \tan^{-1}\left(-\frac{24}{7}\right)\). Calculate this value using a calculator.

Step 4: Calculate θ

Once you have \(2\theta\), divide by 2 to find \(\theta\). So \(\theta = \frac{1}{2} \cdot \tan^{-1}\left(-\frac{24}{7}\right)\).

Step 5: Calculate \(\sin \theta\) and \(\cos \theta\)

Using \(\theta\), compute \(\sin \theta\) and \(\cos \theta\). This can be done using a calculator or trigonometric identities if the angle is known.

Key Concepts

Conic Sections

Conic sections are curves obtained by the intersection of a plane with a cone. These curves can include circles, ellipses, parabolas, and hyperbolas. Each type of conic section has unique properties and equations.
When a quadratic equation contains a cross term or mixed term such as `xy`, it often signifies the presence of a rotated conic. In the context of conic sections, eliminating the `xy` term through rotation can help in easier graphing and identification of the conic shape.
The general form of a conic section equation is:

• \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \)

Understanding conic sections is crucial for solving equations with rotation of axes, as it makes simplifying the equation's form possible.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. These identities are vital tools in simplifying expressions and solving trigonometric problems.
Some basic trigonometric identities that you might already know include:

• Pythagorean identities:
  • \( \sin^2\theta + \cos^2\theta = 1 \)
  • \( 1 + \tan^2\theta = \sec^2\theta \)
• Angle sum and difference identities:
  • \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \)
  • \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \)

These identities are especially useful when calculating trigonometric functions for a rotated angle, such as in the context of the exercise where rotation is required to eliminate the `xy` term.

Tangent Function

The tangent function, denoted by \( \tan\theta \), is one of the basic trigonometric functions. It is defined as the ratio of the sine and cosine functions:

• \( \tan\theta = \frac{\sin\theta}{\cos\theta} \)

It describes the steepness or slope of a line associated with a given angle, often used in the context of rotations.
In scenarios involving rotation of axes, especially to eliminate mixed terms like `xy`, the tangent function plays a crucial role. By solving \( \tan2\theta = \frac{B}{A-C} \), we can determine the amount of rotation needed to simplify the conic equation.
This formula helps us find the new orientation of the conic by first finding \( 2\theta \), and then subsequently \( \theta \), the actual angle of rotation. Understanding how \( \tan2\theta \) is used is key in mastering the rotation of axes.