Problem 12 - 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. $$\boldsymbol{\theta}=60^{\circ},(3,1)$$

Step-by-Step Solution

Answer

The new coordinates are given by (\(x^{\prime}\), \(y^{\prime}\))

Step 1: Identify Given Values

The angle of rotation, \(\theta\), is given as \(60^{\circ}\), which must be converted to radians for calculation. The original coordinates in the \(x y\) system are \(3,1\).

Step 2: Convert Degrees to Radians

As trigonometric functions in calculators typically use radian measures, convert \(\theta\) from degrees to radians using the relation \(1^{\circ} = \pi / 180\) radians. Hence, \(\theta = 60 \times (\pi / 180) = \pi / 3\) radians.

Step 3: Apply the Rotation Matrix

Substitute \(\theta = \pi / 3\) and the coordinates into the rotation matrix \[ \begin{{bmatrix}} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \ \end{{bmatrix}} \begin{{bmatrix}} x \ y \ \end{{bmatrix}} \] leading to new coordinates (\(x^{\prime}\), \(y^{\prime}\))

Key Concepts

Rotation Matrix

The concept of a rotation matrix is at the core of transforming coordinates in a 2D plane. A rotation matrix is a specific kind of transformation matrix used to rotate points around the origin in a coordinate system. It is a 2x2 matrix that utilizes trigonometric functions cosine and sine to achieve rotation.

When dealing with rotations, the rotation matrix for a given angle \( \theta \) is defined as:

• First row: \( \cos \theta, -\sin \theta \)
• Second row: \( \sin \theta, \cos \theta \)

To rotate a point given by \((x, y)\) in the original coordinate system, you multiply the rotation matrix with the coordinates vector:\[ \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \]This multiplication yields the new coordinates \((x', y')\), representing the position of the point in the rotated system. Understanding how to construct and use a rotation matrix is essential for performing these coordinate transformations effectively.

Radians Conversion

When dealing with angles in mathematics, especially in trigonometry, it is integral to be comfortable with converting between degrees and radians. Calculators usually require angles to be in radians when evaluating trigonometric functions.

• The conversion from degrees to radians uses the fact that \(180^{\circ} = \pi\) radians.
• Thus, to convert an angle \(\theta\) in degrees to radians, you multiply by \(\pi/180\).

For example, converting \(60^{\circ}\) to radians is done by:\[ 60 \times \frac{\pi}{180} = \frac{\pi}{3} \]This conversion is crucial when using a rotation matrix, as it ensures that the input for the trigonometric functions corresponds to their expected format. Mastering radians conversion is a critical step in solving problems involving angle rotations.

Trigonometric Functions

Trigonometric functions like sine and cosine play an essential role in coordinate rotations. They help determine how much a point in a 2D plane should move along the x-axis and y-axis when rotated.

Here’s how cosine and sine operate in a rotation matrix:

• Cosine (\(\cos\)): Represents the x-component of a point’s position. It helps maintain the point’s distance from the origin along the x-direction.
• Sine (\(\sin\)): Moves the point along the y-direction, shifting its position perpendicular to the x-component.

In the rotation matrix:

• \(\cos \theta\) is used twice — in both the horizontal and vertical reshuffling of coordinates — it adjusts both x and y components.
• \(-\sin \theta\) is used to rotate the y-component in the opposite direction.
• \(\sin \theta\) adds a lateral shift to the x-component.

Understanding how these trigonometric functions influence rotation will greatly improve comprehension of coordinate transformations and enhance your problem-solving skills.