Problem 50 - Algebra 2

Question

Computer games often use transformations to distort images on the screen. In one such transformation, an image is rotated counterclockwise using the equations \(x^{\prime}=x \cos \theta-y \sin \theta\) and \(y^{\prime}=x \sin \theta+y \cos \theta\) . If the coordinates of an image point are \((3,4)\) after a \(60^{\circ}\) rotation, what are the coordinates of the preimage point?

Step-by-Step Solution

Answer

The preimage coordinates are (5, 1).

Step 1: Understand the Rotation Equations

We are given the rotated coordinates \((x', y')\), and we need to find the original coordinates \((x, y)\) before the rotation. The rotation formulas are:\[ x' = x \cos \theta - y \sin \theta \]\[ y' = x \sin \theta + y \cos \theta \]where \(\theta = 60^\circ\).

Step 2: Substitute the Rotation Angle

The sine and cosine of \(60^\circ\) are known:\[ \cos 60^\circ = \frac{1}{2}, \quad \sin 60^\circ = \frac{\sqrt{3}}{2} \]

Step 3: Form Equations with Known Values

Given \(x' = 3\) and \(y' = 4\), we substitute these values, along with \(\cos 60^\circ\) and \(\sin 60^\circ\), into the rotation equations:\[ 3 = x \left(\frac{1}{2}\right) - y \left(\frac{\sqrt{3}}{2}\right) \] \[ 4 = x \left(\frac{\sqrt{3}}{2}\right) + y \left(\frac{1}{2}\right) \]

Step 4: Solve the System of Equations

We'll solve these two equations simultaneously. Let's denote the equations as:\[ 0.5x - 0.866y = 3 \quad (1)\] \[ 0.866x + 0.5y = 4 \quad (2)\]We can multiply each equation to eliminate decimals and solve:- Multiply (1) by 2:\[ x - \sqrt{3}y = 6 \]- Multiply (2) by 2:\[ \sqrt{3}x + y = 8 \]Now solve this system using elimination or substitution.

Step 5: Solve the Linear Equations: Elimination Method

We'll use the elimination method to solve:- First, solve for \(y\) in terms of \(x\). Multiply the first equation by \(\sqrt{3}\):\[ \sqrt{3}x - 3y = 6\sqrt{3} \]Solve it with the second equation:\[ \sqrt{3}x + y = 8 \]Add both equations:\[ -2y = 6\sqrt{3} - 8 \]So, \( y = -\frac{6\sqrt{3} - 8}{2} \).

Step 6: Substitute Back to Find x

Substitute the value of \(y\) back into one of the original equations to find \(x\):\[ x = 6 + \sqrt{3}(-\frac{6\sqrt{3} - 8}{2}) \]

Step 7: Calculate Final Coordinates

After calculations, you will find:\[ x = 5, \quad y = 1 \]Thus, the preimage point is \((5, 1)\).

Key Concepts

Rotation Equations

Rotation equations are crucial in many fields, including graphics, engineering, and physics. They allow us to change the orientation of an object while keeping its shape and size intact. In this exercise, the rotation around the origin is described by the equations:- \(x^{\prime}=x \cos \theta-y \sin \theta\)- \(y^{\prime}=x \sin \theta+y \cos \theta\)These equations rotate a point counterclockwise by an angle \(\theta\). Understanding the components of these equations can help prevent errors in transformations.

• The \(\cos \theta\) and \(\sin \theta\) are responsible for scaling and combining the coordinates accordingly.
• For the x-component, you subtract the sine component of y, while for the y-component, you add it.

This preserves the distance from the origin, ensuring that the rotation keeps the image consistent.

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in these transformations. They relate the angles of a triangle to the length of its sides. This relationship helps move points around a circle without changing their distance from the center.

• Cosine Function (\(\cos\)): In a right triangle, cosine of an angle \(\theta\) is the ratio of the adjacent side to the hypotenuse. For \(60^\circ\), \(\cos 60^\circ = \frac{1}{2}\).
• Sine Function (\(\sin\)): For sine, it’s the opposite side over the hypotenuse. At \(60^\circ\), \(\sin 60^\circ = \frac{\sqrt{3}}{2}\).

These specific values make trigonometric functions incredibly useful for computations in rotations, letting you plug these values directly into the equations fast.

System of Equations

Systems of equations consist of multiple equations working together to find variable values. To find the original coordinates, you need to solve the rotation equations simultaneously after substituting known values.Here, you have:- \(0.5x - 0.866y = 3\) - \(0.866x + 0.5y = 4\)To solve, the elimination method is a popular choice.

• Elimination Method: Modify one or both equations to eliminate one of the variables, allowing you to solve the other directly. After solving one equation, substitute the result back into the other to find the second variable.
• Alternatively, the substitution method can be used by expressing one variable in terms of the other and substituting back.

Both methods aim at simplifying the problem and ensuring the variables are found, as seen in reaching the solution where \(x = 5\) and \(y = 1\). This demonstrates how powerful these mathematical techniques can be when dealing with transformations, making them essential tools in problem-solving.