Problem 7 - Precalculus: Mathematics for Calculus

Question

Determine the XY-coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(0,2), \quad \phi=55^{\circ}$$

Step-by-Step Solution

Answer

The new coordinates are approximately (1.6384, 1.1472).

Step 1: Identify the Rotation Formula

To find the new coordinates after a rotation through angle \( \phi = 55^{\circ} \), use the transformation formulas: \( x' = x \cos \phi + y \sin \phi \) and \( y' = -x \sin \phi + y \cos \phi \). Here \( x = 0 \) and \( y = 2 \).

Step 2: Substitute Original Coordinates

Substitute \( x = 0 \) and \( y = 2 \) into the formulas: \( x' = 0 \cdot \cos 55^{\circ} + 2 \cdot \sin 55^{\circ} \) and \( y' = -0 \cdot \sin 55^{\circ} + 2 \cdot \cos 55^{\circ} \).

Step 3: Calculate New \( x' \) Coordinate

Calculate \( x' = 2 \cdot \sin 55^{\circ} \). Using \( \sin 55^{\circ} \approx 0.8192 \), we find \( x' \approx 2 \times 0.8192 = 1.6384 \).

Step 4: Calculate New \( y' \) Coordinate

Calculate \( y' = 2 \cdot \cos 55^{\circ} \). Using \( \cos 55^{\circ} \approx 0.5736 \), we find \( y' \approx 2 \times 0.5736 = 1.1472 \).

Step 5: Provide Final Coordinates

The new coordinates after the rotation are approximately \( (1.6384, 1.1472) \).

Key Concepts

Trigonometric Functions

Trigonometric functions play a crucial role in determining the new coordinates of a point after rotation. Specifically, when dealing with coordinate rotation, the sine and cosine functions are essential.

These functions help understand the changes in the position of the points when an angle is applied to them.

• Sine (\( \sin \phi \)): Represents the y-coordinate of the point on the unit circle that corresponds to the angle \( \phi \).
• Cosine (\( \cos \phi \)): Represents the x-coordinate of the point on the unit circle that corresponds to angle \( \phi \).

In coordinate rotation problems, these functions are used to systematically transform the original coordinates to their new, rotated positions by applying the transformation formulas.

It is important to understand that \( \sin \phi \) and \( \cos \phi \) are used because they maintain the length of the radius in the unit circle, ensuring the point's position relative to the origin remains consistent in terms of magnitude.

Transformation Formulas

When performing coordinate rotations, transformation formulas are key to converting the original coordinates into a new frame of reference. These are derived from the properties of circles and angles.

The transformation formulas for a counterclockwise rotation by an angle \( \phi \) are:

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

These formulas work by projecting the original coordinates' contributions on the new axes formed by the angle of rotation. The idea is to transform the point's position from one orientation to another smoothly.

By substituting the original coordinates into these formulas, you can compute the resulting coordinates after the indicated rotation, generating new points that maintain the distance from the origin, highlighting their utility in varied mathematical and engineering applications.

XY-Coordinates

Whenever dealing with coordinate geometry, understanding the concept of XY-coordinates is fundamental. An XY-coordinate system, also known as the Cartesian coordinate system, allows us to plot points by defining positions along the horizontal (x-axis) and the vertical (y-axis) dimensions.

In the context of coordinate rotation:

• The original coordinates are written as \((x, y)\), indicating their exact position in the 2D plane.
• After applying the transformation formulas, new coordinates \((x', y')\) represent their rotated positions.

This transformation does not alter the relative distances between points but instead provides a new orientation to view the same graphic data.

Coordinate systems enable the analysis and graphical representation of spatial relationships in mathematics, leading to deeper insights into geometrical properties and inter-relationships.