Problem 4 - Algebra and Trigonometry

Question

Write each equation in terms of a rotated \(x^{\prime} y^{\prime}-\)system using \(\theta,\) the angle of rotation. Write the equation involving \(x^{\prime}\) and \(y^{\prime}\) in standard form. $$13 x^{2}-10 x y+13 y^{2}-72=0 ; \theta=45^{\circ}$$

Step-by-Step Solution

Answer

The final equation in terms of the rotated \(x^{\prime}\) and \(y^{\prime}\) coordinates will be represented as \(A (x^{\prime})^2 + B (y^{\prime})^2 = 1\).

Step 1: Rotate the Coordinate System

Find the rotated coordinates \(x^{\prime}\) and \(y^{\prime}\) using the rotation matrix for \(\theta = 45^{\circ}\). This is done using the formulas: \(x^{\prime} = x\cos\theta - y\sin\theta\) and \(y^{\prime} = x\sin\theta +y\cos\theta\). Fill in for \(\theta = 45^{\circ}\). Instead of degrees, use radians, where \(45^{\circ}\) is equal to \(\pi / 4\). Thus, the transformation will be: \(x = x^{\prime}\cos(\pi/4) + y^{\prime}\sin(\pi/4)\) and \(y = -x^{\prime}\sin(\pi/4) + y^{\prime}\cos(\pi/4)\).

Step 2: Substitute coordinates into the given equation

Substitute the results from step 1 into the original equation and simplify: \(13(x^{\prime}\cos(\pi/4) + y^{\prime}\sin(\pi/4))^{2} - 10(x^{\prime}\cos(\pi/4) + y^{\prime}\sin(\pi/4))(-x^{\prime}\sin(\pi/4) + y^{\prime}\cos(\pi/4)) + 13(-x^{\prime}\sin(\pi/4) + y^{\prime}\cos(\pi/4))^{2} - 72 = 0\) .

Step 3: Simplify the equation to standard form

Simplify this equation to obtain the equation in terms of \(x^{\prime}\) and \(y^{\prime}\). This will be obtained by multiplying out each term in the equation from step 2 and simplifying to arrive at an equation of the form \(A (x^{\prime})^2 + B (y^{\prime})^2 = 1\).

Key Concepts

Coordinate Transformation

Transforming coordinates involves changing how we describe the positions of points. In many mathematical applications, it's useful to rotate the coordinate system by a certain angle, especially when dealing with conic sections or other geometrical figures. This process can simplify the complexity of the equations we are working with.
For our problem, we're dealing with a rotation angle \(\theta = 45^{\circ}\), which is equivalent to \(\frac{\pi}{4}\) radians. By rotating the coordinate system, we transform standard \(x,y\) coordinates into a new \(x',y'\) system using the rotation matrix.

Standard Form Equation

After transforming the coordinates, an important step is to express the transformed equation in a standard form. The standard form makes it easier to identify the type of conic section represented by the equation, like circles, ellipses, or hyperbolas.

• For example, an equation of the form \(A (x')^2 + B (y')^2 = 1\) can represent an ellipse.
• The process involves substituting the new expressions for \(x\) and \(y\) in terms of \(x'\) and \(y'\) into the original equation, and then simplifying it to fit the standard form.

Working in standard form provides clarity and is often required for further calculations or graphing.

Rotation Matrix

The rotation matrix is a key tool for transforming coordinates. It's used to calculate the new positions \(x', y'\) after rotating the axes by a specified angle \(\theta\).
The formulas: \(x' = x\cos\theta - y\sin\theta\) and \(y' = x\sin\theta + y\cos\theta\) come from this matrix, and they allow us to rotate any point around the origin:

• The rotation matrix for an angle \(\theta\) is given by:\[\begin{pmatrix}\cos \theta & -\sin \theta \\sin \theta & \cos \theta\end{pmatrix}\]
• By applying this matrix, we derive the new coordinates from the original ones.

This method can be applied to any rotational transformation.

Radian Measure

Angles can be measured in degrees or radians. Radians are favored in many mathematical applications because they simplify the mathematics involved, particularly in calculus and trigonometry.
One full rotation around a circle is \(2\pi\) radians, equivalent to \(360^{\circ}\). Thus, converting from degrees to radians is done using the relation:

• \(\theta_{\text{radians}} = \frac{\pi}{180} \times \theta_{\text{degrees}}\).
• For example, \(45^{\circ} = \frac{\pi}{4}\) radian.

Working in radians can make calculations involving trig functions more straightforward, and is a preferred unit in most mathematical and engineering fields.