Problem 2 - Precalculus: Mathematical for Calculus

Question

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(-2,1), \quad \phi=30^{\circ}$$

Step-by-Step Solution

Answer

New coordinates: \((-\sqrt{3} + \frac{1}{2}, 1 + \frac{\sqrt{3}}{2})\).

Step 1: Identify the original coordinates and angle of rotation

The original coordinates of the point are \((-2, 1)\), and the angle \(\phi\) through which the axes are rotated is \(30^{\circ}\).

Step 2: Convert the angle into radians

To perform calculations using trigonometric functions, convert the angle from degrees to radians. The formula to convert degrees to radians is:\[\text{{radians}} = \text{{degrees}} \times \frac{\pi}{180}\]So, \(30^{\circ}\) becomes:\[30^{\circ} \times \frac{\pi}{180} = \frac{\pi}{6}\]

Step 3: Use the rotation formulas to find the new coordinates

The transformation formulas for rotation are:\[x' = x \cos(\phi) + y \sin(\phi)\]\[y' = -x \sin(\phi) + y \cos(\phi)\]Substitute \((-2, 1)\) for \((x, y)\) and \(\frac{\pi}{6}\) for \(\phi\).

Step 4: Calculate \(x'\) using rotation formula

Compute the new x-coordinate:\[x' = -2 \cos\left(\frac{\pi}{6}\right) + 1 \sin\left(\frac{\pi}{6}\right)\]\[\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}, \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\]So,\[x' = -2 \times \frac{\sqrt{3}}{2} + 1 \times \frac{1}{2} = -\sqrt{3} + \frac{1}{2}\]

Step 5: Calculate \(y'\) using rotation formula

Compute the new y-coordinate:\[y' = -(-2) \sin\left(\frac{\pi}{6}\right) + 1 \cos\left(\frac{\pi}{6}\right)\]\[y' = 2 \times \frac{1}{2} + 1 \times \frac{\sqrt{3}}{2} = 1 + \frac{\sqrt{3}}{2}\]

Step 6: Write down the new coordinates

After performing the calculations, the new coordinates of the point after rotation are:\[(-\sqrt{3} + \frac{1}{2}, 1 + \frac{\sqrt{3}}{2})\]

Key Concepts

Understanding Coordinate Rotation

Coordinate rotation is a fundamental concept in mathematics, particularly in coordinate geometry. It involves rotating a point around the origin of a coordinate system by a specified angle. The operation transforms the point's position from its original coordinates to new ones in the rotated frame. The process is akin to turning a flashlight; the light beam is the original points, and the angle of rotation adjusts its position. This concept is essential in various fields such as physics, computer graphics, and engineering. To correctly perform coordinate rotation, it is crucial to follow a structured approach:

• Identify the original coordinates and the angle of rotation.
• Transform the angle into radians, which is necessary for applying trigonometric functions.
• Apply mathematical formulas to compute the new position.

Understanding these steps ensures accurate transformations in mathematical problems and real-world applications.

Trigonometric Functions in Rotation

Trigonometric functions play an integral role in the process of rotating coordinates. These functions, such as sine and cosine, are used to calculate the new positions of points after a rotation. When dealing with a rotation, the cosine of the angle helps determine how much of the original x-coordinate contributes to the new x-coordinate. Conversely, the sine function measures the y-coordinate's contribution. It is crucial to remember:

• The cosine function goes with the original x-values to affect the rotated x-coordinate.
• The sine function goes with the original y-values for the y-coordinate transformation.
• Sometimes, the sine function also contributes to x values, but with a negative sign, and vice versa for y-coordinates in the formulas.

Any slight misstep in using these functions can lead you to an incorrect result, so precision is key.

Radian Conversion Simplified

Converting angles from degrees to radians is a vital step in mathematical calculations involving trigonometric functions. In many mathematical computations, particularly those involving trigonometric functions, angles are conventionally expressed in radians rather than degrees. Here’s a quick refresher on how to do it:

• Use the conversion formula: \[ ext{{radians}} = ext{{degrees}} \times \frac{\pi}{180}\]
• Simply multiply your degree measurement by the fraction \(\frac{\pi}{180}\) to get the angle in radians.

In our problem, to convert \(30^{\circ}\) into radians, multiply \(30\) by \(\frac{\pi}{180}\), resulting in \(\frac{\pi}{6}\). This conversion allows you to proceed with trigonometric calculations correctly, as trigonometric functions in most calculators and mathematical formulas use radians.

Applying Rotation Formulas

Rotation formulas are the key tools for finding new coordinates after a rotation. They are derived using trigonometric identities, considering how x and y components change with rotation.The fundamental formulas are:

• For the new x-coordinate: \(x' = x \cos(\phi) + y \sin(\phi)\)
• For the new y-coordinate: \(y' = -x \sin(\phi) + y \cos(\phi)\)

By substituting the given point and converted angle into these formulas, you calculate the transformed position.For example, substituting \((-2, 1)\) and \(\phi = \frac{\pi}{6}\):

• For \(x'\): Combine \(-2\cos(\frac{\pi}{6})\) and \(1\sin(\frac{\pi}{6})\) to get \(-\sqrt{3} + \frac{1}{2}\).
• For \(y'\): Sum \(2\sin(\frac{\pi}{6})\) and \(1\cos(\frac{\pi}{6})\) to find \(1 + \frac{\sqrt{3}}{2}\).

This process translates complex movements into manageable mathematics, allowing us to understand and predict the behavior of rotated elements.