Problem 3 - Precalculus: Mathematics for Calculus

Question

Determine the XY-coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(1,1), \quad \phi=45^{\circ}$$

Step-by-Step Solution

Answer

The new coordinates are (0, \(\sqrt{2}\)).

Step 1: Understand the Rotation Formula

To find the new coordinates of a point after a rotation, we use the rotation transformation formulas: \[ x' = x \cdot \cos(\phi) - y \cdot \sin(\phi) \]\[ y' = x \cdot \sin(\phi) + y \cdot \cos(\phi) \]where \((x, y)\) are the original coordinates, \((x', y')\) are the new coordinates, and \(\phi\) is the angle of rotation.

Step 2: Substitute Known Values into the Formulas

For this problem, the point is \((x, y) = (1, 1)\) and the rotation angle \(\phi = 45^{\circ}\). We substitute these values into the formulas:\[ x' = 1 \cdot \cos(45^{\circ}) - 1 \cdot \sin(45^{\circ}) \]\[ y' = 1 \cdot \sin(45^{\circ}) + 1 \cdot \cos(45^{\circ}) \].

Step 3: Calculate Trigonometric Values

For \(\phi = 45^{\circ}\), we have \(\cos(45^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\). Substitute these values into the equations:\[ x' = 1 \times \frac{\sqrt{2}}{2} - 1 \times \frac{\sqrt{2}}{2} \]\[ y' = 1 \times \frac{\sqrt{2}}{2} + 1 \times \frac{\sqrt{2}}{2} \].

Step 4: Simplify the Equations

Simplify both of the new coordinate formulas:\[ x' = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 0 \]\[ y' = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2} \].

Step 5: Write the New Coordinates

The new coordinates of the point after the rotation of \(45^{\circ}\) are \((x', y') = (0, \sqrt{2})\).

Key Concepts

Rotation Transformation Formulas

To rotate a point in the coordinate plane, we use the essential rotation transformation formulas. These equations help in finding the new position of a point when the axes are rotated by a specific angle. The formulas are:

• \( x' = x \cdot \cos(\phi) - y \cdot \sin(\phi) \)
• \( y' = x \cdot \sin(\phi) + y \cdot \cos(\phi) \)

In this context:- \((x, y)\) represents the original coordinates of the point.- \((x', y')\) are the new coordinates post-rotation.- \(\phi\) denotes the angle of rotation.
These formulas adjust the original coordinates according to the rotation angle, ensuring the point is correctly repositioned in the new orientation.

Trigonometric Values

Trigonometric values are key components in calculating rotated coordinates. For common angles, such as \(45^{\circ}\), these values are often used repeatedly and are easily memorized. For \(\phi = 45^{\circ}\), the values are:

• \(\cos(45^{\circ}) = \frac{\sqrt{2}}{2}\)
• \(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\)

These trigonometric ratios represent the cosine and sine of the angle and determine how much the x and y coordinates are scaled during rotation. Their importance lies in the fact that they effectively scale the contributions from each original coordinate, without which a mathematically correct rotation would be impossible.

Rotation Angle

The rotation angle \(\phi\) is a pivotal factor in determining how the coordinates of the point will transform. It influences both the direction and the magnitude of the rotation. For example, an angle of \(45^{\circ}\) results in a precise rotation that is midway between aligning completely with the x or y axis.
Choosing an appropriate angle depends on the goal of the rotation. Some practical applications include aligning objects, correcting perspective, or rotating graphical elements for design layouts. The process of using specific angles follows naturally in these contexts, highlighting its application end goals.

XY-Coordinates

XY-coordinates are the backbone of the coordinate system, providing a system to map out the position of points in 2-D space. Initially given as \((1,1)\), they represent both position and orientation before rotation occurs.
When rotated, these coordinates transform. For instance, through a \(45^{\circ}\) rotation, the new coordinates become \((0, \sqrt{2})\). This result symbolizes how the point moves along the plane while maintaining its distance from the origin.
Understanding the role of these coordinates is crucial for analyzing the impact of rotations, whether for geometric transformations or for dynamic systems involving movement.