Problem 8 - Precalculus: Mathematics for Calculus

Question

Determine the XY-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 rotated point coordinates are (5, 3).

Step 1: Understand the Rotation Formula

When rotating a point \((x, y)\) by an angle \(\phi\) around the origin, the new coordinates \((x', y')\) are given by: \[ x' = x \cos \phi + y \sin \phi \] \[ y' = -x \sin \phi + y \cos \phi \] We will use this formula to find the new coordinates of the point \((\sqrt{2}, 4\sqrt{2})\) when \(\phi = 45^\circ\).

Step 2: Substitute the Given Values into the Formula

First, substitute \(x = \sqrt{2}\), \(y = 4\sqrt{2}\), and \(\phi = 45^\circ\) into the rotation formulas. The trigonometric values for \(45^\circ\) are \(\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}\).

Step 3: Calculate New X-Coordinate

Substitute into the formula for the new x-coordinate: \[ x' = \sqrt{2} \cdot \frac{\sqrt{2}}{2} + 4\sqrt{2} \cdot \frac{\sqrt{2}}{2} \] Simplify: \[ x' = \frac{2}{2} + \frac{8}{2} \] \[ x' = 5 \]

Step 4: Calculate New Y-Coordinate

Substitute into the formula for the new y-coordinate: \[ y' = -\sqrt{2} \cdot \frac{\sqrt{2}}{2} + 4\sqrt{2} \cdot \frac{\sqrt{2}}{2} \] Simplify: \[ y' = -\frac{2}{2} + \frac{8}{2} \] \[ y' = 3 \]

Step 5: Final Answer

The new coordinates of the point after rotation are \((5, 3)\).

Key Concepts

Trigonometric Functions

Trigonometric functions play a crucial role in coordinate rotation. When we rotate a point in the coordinate plane, we utilize the sine and cosine functions. These functions help us determine the new position of any point after rotation from the origin.

• Cosine (\( \cos \phi \)): Represents the adjacent side of the angle when \(\phi\) is the angle of rotation. In our problem, when \(\phi = 45^\circ\), \(\cos 45^\circ = \frac{\sqrt{2}}{2}\).
• Sine (\( \sin \phi \)): Represents the opposite side related to the angle of rotation. For the same angle \(\phi = 45^\circ\), \(\sin 45^\circ = \frac{\sqrt{2}}{2}\).

After calculating these functions, they are incorporated into the formulas that govern the rotation. As \(\phi = 45^\circ\), the symmetry simplifies calculations, making both \(\cos\) and \(\sin\) equal. This unique situation makes the problem easier to solve, as both functions have the same value.

Coordinate Transformation

A coordinate transformation involves changing the basis of a coordinate system. When performed, it adjusts the original coordinates, often to make operations such as rotation more straightforward. In our exercise, the point \((\sqrt{2}, 4\sqrt{2})\) is transformed based on a new angle \(\phi\).

The transformation can be outlined in two primary steps:

• Identify Original Coordinates: This involves determining the initial position of the point before any rotation or transformation—in this case, \(x = \sqrt{2}\) and \(y = 4\sqrt{2}\).
• Apply the Rotation Formula: Utilize the established formulas, input the trigonometric functions for the angle given, and solve. Coordinating both equations gives a holistic transformation: \(x' = x \cos \phi + y \sin \phi\) and \(y' = -x \sin \phi + y \cos \phi\).

Transformations like these make analyzing points in altered geometric conditions much more systematic and manageable.

Rotation Formula

The rotation formula is critical for moving points in rotational motion around the Cartesian coordinate plane. This formula helps derive the new set of coordinates for any given point, based on a specific rotation angle \(\phi\).

• Components of the Formula: The formulas used for this transformation are:\[x' = x \cos \phi + y \sin \phi\]\[y' = -x \sin \phi + y \cos \phi\]These formulas effectively convert the position by incorporating trigonometric identities.
• Application Example: Our exercise utilized these formulas, where the given point \((\sqrt{2}, 4\sqrt{2})\) was subjected to a \(45^\circ\) rotation. For \(x'\), it involved multiplying each coordinate with the cosine and sine functions appropriately. Similarly, \(y'\) was achieved through a similar approach with a slightly altered calculation due to the negative sine factor.

The precision of these calculations highlights how important understanding and applying the right formulas are. The process enabled the conversion of the point to its new position after rotation, providing the final result of \((5, 3)\).