Problem 5 - 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. \(\theta=90^{\circ}, (0, 3)\)

Step-by-Step Solution

Answer

The coordinates of the point in the rotated coordinate system are (-3,0).

Step 1: Insert values into the rotation formulas

In the given problem, the rotation angle \(\theta\) equals 90 degrees, and the point in the xy system being transformed has coordinates (0,3). Thus, the equations will look like this: \(x'=0\cos{90}-3\sin{90}\) and \(y'=0\sin{90}+3\cos{90}\).

Step 2: Simplify the equations

Using basic trigonometric knowledge it can be seen that \(\cos{90}=0\) and \(\sin{90}=1\). Now, the formulas simplify to \(x'=-3\) and \(y'=0\).

Step 3: Confirm the result

The equations now read \(x'=-3\) and \(y'=0\). Thus, the point (0,3) with respect to the original coordinate system gets transformed to (-3,0) with respect to the new, rotated coordinate system.

Key Concepts

Trigonometric Rotation Formulas

Understanding the trigonometric rotation formulas is essential for analyzing how points transform when we rotate the coordinate system. When we talk about rotating a coordinate system by an angle \theta, we're describing a transformation that changes the original coordinates of a point to new coordinates based on that rotation.

The basic rotation formulas for rotating a point (x, y) counterclockwise around the origin by an angle \theta are given by:

• For the x-coordinate: \(x' = xcos(\theta) - ycos(\theta)\)
• For the y-coordinate: \(y' = xcos(\theta) + ycos(\theta)\)

These equations use cosine and sine, which are fundamental trigonometric functions representing the projections of a unit radius on the axes as an angle sweeps from the positive x-axis. In the context of our exercise, the rotation angle is 90 degrees, a particular case where cosine of 90 degrees vanishes and sine of 90 degrees reaches its maximum, being equal to 1. This leads to a very simplified set of formulas for the transformation: \(x' = -y\) and \(y' = x\) after substituting the respective cosine and sine values.

Transformation of Coordinates

The transformation of coordinates is a mathematical operation applied to points when changing between different coordinate systems or reference frames. This procedure is central to fields like physics, engineering, and computer graphics, where the relationship and positions between objects in space are crucial.

In the case of rotating coordinate systems, transformation becomes a matter of finding how each point's coordinates are redefined. It is not just about crunching the numbers; it's about visualizing the space being transformed and understanding that each point maintains its original position in space, while the grid, or reference frame, shifts around it.

For instance, when given a point (0, 3) and asking for its coordinates after a 90-degree rotation, as in our exercise, we are not moving the point itself but the grid it is on. By applying the trigonometric rotation formulas, we can find the new coordinates of the point within this rotated grid, which can help us in solving practical problems such as navigating a robot in a new direction or rotating objects in a video game.

Rotated Coordinate System

A rotated coordinate system results from turning the axes of the original coordinate system about the origin (0,0). This rotation changes the frame of reference for points within this system. When the system is rotated, every point's coordinates transform in a way that preserves their spatial relationships within the system.

In our exercise example, rotating the coordinate system by 90 degrees counterclockwise has visually moved the point (0, 3) to a new position relative to the new axes. However, physically, the point is stationary, and only the grid has moved. This distinction is critical to avoid confusion and to understand that rotations do not affect the points themselves but the reference frame through which we view or measure them. Understanding a rotated coordinate system is fundamental in navigation, graphics design, and many branches of science and technology where orientation matters.