Problem 8 - Precalculus

Question

The \(x^{\prime} y^{\prime}\) -coordinate system has been rotated \(\theta\) degrees from the \(x y\) -coordinate system. The coordinates of a point in the \(x y\) -coordinate system are given. Find the coordinates of the point in the rotated coordinate system. $$\theta=30^{\circ},(2,4)$$

Step-by-Step Solution

Answer

The coordinates of the point in the rotated system are approximately (1.732, 3.964).

Step 1: Conversion of Degrees to Radians

The rotation angle \( \theta \) is given in degrees. Convert this to radians, as the standard mathematical functions in maths use radian measures. The conversion can be done using the relation: 1 degree = \( \pi / 180 \) radians. Using this, we get \( \theta = 30^{\circ} = 30 \cdot \pi / 180 = \pi / 6 \) radians.

Step 2: Setup Rotation Matrix

Use the standard formula for rotation in a two-dimensional space. A point (x, y) can be rotated around the origin by the angle \( \theta \) using the following rotation matrix: \( [[\cos(\theta), -\sin(\theta)], [\sin(\theta), \cos(\theta)]] \). Substitute \( \theta = \pi / 6 \) into this matrix to get: \( [[\cos(\pi / 6), -\sin(\pi / 6)], [\sin(\pi / 6), \cos(\pi / 6)]] \)

Step 3: Apply the Rotation Matrix

To find the new coordinates (x', y'), multiply the original coordinates (2,4) with the rotation matrix. Perform matrix multiplication to obtain: x' = 2 * cos(\pi / 6) - 4 * sin(\pi / 6) = 1.732 and y' = 2 * sin(\pi / 6) + 4 * cos(\pi / 6) = 3.964.

Key Concepts

Understanding the Rotation Matrix

A rotation matrix is a mathematical tool used to rotate points in a coordinate space. Imagine a point on a flat surface. When we wish to "turn" this point around a central pivot, akin to rotating a wheel on an axle, we apply a rotation matrix. The matrix gives the new position of the point after the desired rotation.

In the two-dimensional space, the rotation matrix is a 2x2 matrix, represented as:

• For a given angle \( \theta \), the matrix is \( \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix} \).
• The angle \( \theta \) is the rotation angle measured from the original axis, typically in radians.

Using trigonometric functions, cosine and sine, this matrix ensures the preserved distance from the point to the origin, meaning the lengths remain constant as the shape rotates. This matrix is essential because it provides a systematic way to change an object's orientation while maintaining its size and shape.

Converting Degrees to Radians

Degrees and radians are two ways of measuring angles. In everyday language, degrees might be more familiar, but in mathematics, particularly calculus and trigonometry, radians are preferred.

To convert an angle from degrees to radians, use the formula:

• \( 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \).

Let's take 30 degrees as an example:

• Multiply 30 by \( \frac{\pi}{180} \) to get \( \frac{\pi}{6} \text{ radians} \).

Using radians simplifies many mathematical equations and calculations since they are based on the properties of circles. It makes computations involving trigonometric functions more consistent, especially when these results are applied in programming and engineering tasks.

The Process of Matrix Multiplication

Matrix multiplication is a method of multiplying two matrices, which represents actions like transformations, rotations, or scaling.

When performing matrix multiplication from a 2x2 matrix and a column matrix for coordinates, follow these steps:

• Take each row of the first matrix and multiply it by each column of the second matrix.
• Combine these products through addition.

For instance, let's consider multiplying a rotation matrix by a point's coordinates:

• The matrix: \( [[\cos(\theta), -\sin(\theta)], [\sin(\theta), \cos(\theta)]] \).
• The coordinates: \( (x, y) \).

Perform the multiplication as follows:

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

This operation results in the new rotated coordinates \( (x', y') \). Matrix multiplication might sound complex at first, but it systematically turns challenging transformations into simpler tasks. Familiarity with the process is essential for those looking to understand geometrical and linear transformations.