Problem 11 - Precalculus with Limits

Question

In Exercises 5-12, the \(x'y'\)-coordinate system has been rotated \(\theta\) degrees from the \(xy\)-coordinate system. The coordinates of a point in the \(xy\)-coordinate system are given. Find the coordinates of the point in the rotated coordinate system. \(\theta=60^{\circ}, (1, 2)\)

Step-by-Step Solution

Answer

The coordinates of the given point in the rotated coordinate system are approximately (-0.37, 2.23)

Step 1: Understand Coordinate Rotation

Coordinate rotation is a sort of transformation that takes points from one coordinate system and translates them into a new coordinate system that's been rotated by a certain angle about the origin. To do this, the rotation transformation uses the following equations:\(x' = x \cosθ - y \sinθ\)\(y' = x \sinθ + y \cosθ\)where \(x'\) and \(y'\) are the coordinates of the point in the new (rotated) system, \(x\) and \(y\) are the original coordinates, and \(\theta\) is the angle of rotation. Now, substituting the values from the given problem into these equations.

Step 2: Substituting Values

Given: Original Point (x, y) => (1, 2) Rotation Angle \(\theta\) => 60°, but we need to convert it to radians for our equations because the trigonometric functions in these equations are actually defined in terms of radians, not degrees. Remembering that \(π\) radians is equal to 180°, we can convert 60° to radians as follows: \(\theta = 60 * \frac{π}{180} = \frac{π}{3}\) radians. Now, substitute these values into the rotation transformation equations

Step 3: Calculating New Coordinates

With the values substituted, the rotation equations become:\(x' = 1*\cos(\frac{\pi}{3}) - 2*\sin(\frac{\pi}{3})\)\(y' = 1*\sin(\frac{\pi}{3}) + 2*\cos(\frac{\pi}{3})\)Simplify above expressions to find the new coordinates. Note: \(\cos(\frac{\pi}{3}) = 1/2\) and \(\sin(\frac{\pi}{3}) = \sqrt{3}/2\).

Step 4: Simplification

Simplify equations to find the values of \(x'\) and \(y'\):\(x' = 1*(1/2) - 2*(\sqrt{3}/2) = 1/2 - \sqrt{3} = -0.37 (rounded to two decimal places)\)\(y' = 1*(\sqrt{3}/2) + 2*(1/2) = \sqrt{3}/2 + 1 = 2.23 (rounded to two decimal places)\)

Key Concepts

rotation transformation

In geometry, a rotation transformation is a way to rotate points around a fixed point, usually the origin of a coordinate system. Picture a point sitting on a spinning disk that rotates by a certain angle. As the disk spins, the point follows a circular path and moves to a new location.
When we talk about rotation transformation in a coordinate system, we need specific equations to find out where each point moves. For a 2D (two-dimensional) rotation, these equations are:

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

Here, \(x'\) and \(y'\) are the new, or transformed, coordinates, and \(x\) and \(y\) are the original coordinates. The angle \(\theta\) is the angle through which the original system has been rotated. Remember, positive angles indicate counterclockwise rotation in most mathematical contexts.

trigonometric functions

Trigonometric functions are fundamental in mathematics, especially when dealing with rotations. They help us relate angles to the lengths of sides in a right triangle. The primary trigonometric functions used in rotations are sine (\(\sin\)) and cosine (\(\cos\)).
In the context of a rotation transformation:

• The function \(\cos\theta\) is the length of the side adjacent to the angle \(\theta\), divided by the length of the hypotenuse in a right triangle.
• Similarly, \(\sin\theta\) is the length of the side opposite the angle \(\theta\), divided by the hypotenuse.

To compute these functions for specific angles like \(60^\circ\), we often use known values from the unit circle, such as \(\cos(\frac{\pi}{3}) = \frac{1}{2}\) and \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\). These values allow us to accurately perform transformations and map points to their new locations.

coordinate system

A coordinate system is a method for specifying each point uniquely in a plane by a pair of numerical coordinates. These coordinates are typically related to a horizontal axis \(x\) and a vertical axis \(y\). When we rotate this system, it's like spinning the whole grid by a certain angle around the origin. As a result, the point (1, 2) in the original \(xy\)-coordinates transforms into a new location in the \(x'y'\)-coordinates system. The transformation depends on how much you "tilt" or rotate the original axes. In most cases, the rotation matrix helps achieve this, automatically adjusting the display from one view to another.The beauty of the coordinate system transformation is that it maintains the relationships and distances between points, simply offering a different perspective of the entire plane.

angle conversion

To work effectively with rotations, particularly when dealing with trigonometric functions, it is often necessary to convert angles from degrees to radians. This is because most mathematical functions, especially those used in programming and analytics, assume angles are provided in radians.The conversion between degrees and radians follows a simple formula:

• \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)

For example, converting \(60^\circ\) to radians involves multiplying 60 by \(\frac{\pi}{180}\), giving you \(\frac{\pi}{3}\) radians.Understanding the need for angle conversion is crucial when dealing with rotations in different contexts, whether you're coding a computer graphics application or solving a geometry problem. It ensures that all functions work under the same assumptions and produce consistent results.