Problem 8 - Algebra and Trigonometry

Question

\(3-8\) . Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$ (\sqrt{2}, 4 \sqrt{2}), \quad \phi=45^{\circ} $$

Step-by-Step Solution

Answer

The new coordinates are \((-3, 5)\).

Step 1: Understanding the Problem

We need to find new coordinates \((X, Y)\) for a given point \((x, y)\) after the coordinate axes have been rotated by an angle \(\phi=45^{\circ}\). The original coordinates are \((\sqrt{2}, 4\sqrt{2})\).

Step 2: Rotation Formulae Introduction

To find the new coordinates after rotation around the origin, we use the rotation transformation equations: \(X = x\cos \phi - y\sin \phi\) and \(Y = x\sin \phi + y\cos \phi\).

Step 3: Apply the Cosine and Sine Values

Calculate \(\cos 45^{\circ}\) and \(\sin 45^{\circ}\). Both are equal to \(\frac{\sqrt{2}}{2}\).

Step 4: Calculate the New X-coordinate

Substitute \(x = \sqrt{2}\), \(y = 4\sqrt{2}\), \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\) and \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\) in the equation \(X = x\cos \phi - y\sin \phi\): \[ X = \sqrt{2} \cdot \frac{\sqrt{2}}{2} - 4\sqrt{2} \cdot \frac{\sqrt{2}}{2} \]This simplifies to: \[ X = 1 - 4 = -3 \]

Step 5: Calculate the New Y-coordinate

Substitute the same values into the equation \(Y = x\sin \phi + y\cos \phi\):\[ Y = \sqrt{2} \cdot \frac{\sqrt{2}}{2} + 4\sqrt{2} \cdot \frac{\sqrt{2}}{2} \]This simplifies to: \[ Y = 1 + 4 = 5 \]

Step 6: Solution Interpretation

The new coordinates of the point \((X, Y)\) after a rotation by \(45^{\circ}\) are \((-3, 5)\).

Key Concepts

Rotation Transformation

In mathematics, a **Rotation Transformation** involves turning a point around a fixed center, usually the origin of the coordinate system. This transformation maintains the shape and size of the object being rotated, which means all distances between points remain constant. When rotating in a 2D space, this is often done using specific equations that alter the original coordinates of the point in relation to a given angle.

• A rotation is mathematically represented by two main equations: - The new x-coordinate: \(X = x\cos \phi - y\sin \phi\) - The new y-coordinate: \(Y = x\sin \phi + y\cos \phi\)
• The angle \(\phi\) denotes how much the point is rotated. This angle is typically measured in degrees or radians.

Understanding these transformations is crucial to analyzing movements in fields such as physics, engineering, and computer graphics.
When applying a rotation transformation, each point given by \((x, y)\) will shift to a new position denoted by \((X, Y)\). This is achieved without scaling or distorting the original figure, ensuring it remains congruent with its previous state.

Trigonometric Values

**Trigonometric Values** are fundamental in calculating the effects of rotation, using basic sine and cosine functions. These functions help determine the new position of a point after rotation by relating an angle with the lengths of sides in a right-angled triangle.

• The value of \(\cos 45^{\circ}\) is \(\frac{\sqrt{2}}{2}\), and likewise, \(\sin 45^{\circ}\) is also \(\frac{\sqrt{2}}{2}\).
• This consistent value simplifies calculations as it allows easy substitution into the rotation transformation equations.

Knowing these values is essential because they help bridge the gap between angles and linear coordinates in the plane.
For any given angle \(\phi\), trigonometric functions provide the multipliative factors by which the original coordinates are scaled during rotation. Using standardized trigonometric values makes these transformations possible without need for a calculator each time.

New Coordinate Calculation

**New Coordinate Calculation** is the practical application of rotation transformation and trigonometric values to find the new position of a point post-rotation. By substituting known values into the rotation equations, one can calculate new coordinates accurately.

Step-by-step Calcuation:

First, ensure you understand the given original coordinates \((x, y)\) and the rotation angle \(\phi\). In our case:

• Original coordinates: \((\sqrt{2}, 4\sqrt{2})\)
• Rotation angle: \(45^{\circ}\)

Next, use the trigonometric values to substitute into the rotation formulas:

• Substitute into the x-coordinate formula: \[X = \sqrt{2} \cdot \frac{\sqrt{2}}{2} - 4\sqrt{2} \cdot \frac{\sqrt{2}}{2} = 1 - 4 = -3\]
• Substitute into the y-coordinate formula: \[Y = \sqrt{2} \cdot \frac{\sqrt{2}}{2} + 4\sqrt{2} \cdot \frac{\sqrt{2}}{2} = 1 + 4 = 5\]

Understanding and applying these calculations shows the resulting point post-rotation, confirming that through this process, the original coordinates have moved to the new coordinates \((-3, 5)\), while maintaining the shape's integrity in space.