Problem 4 - Precalculus

Question

An angle of rotation is specified, followed by the coordinates of a point in the \(x\) -y system. Find the coordinates of each point with respect to the \(x\) '-y' system. $$\theta=45^{\circ} ;(x, y)=(0,-2)$$

Step-by-Step Solution

Answer

The new coordinates are \((-\sqrt{2}, -\sqrt{2})\).

Step 1: Understand Rotation of Axes

In this problem, we need to rotate the coordinate system by an angle \(\theta = 45^{\circ}\). The given point \((x, y) = (0, -2)\) needs to be recalculated to fit into the new rotated coordinate system \((x', y')\).

Step 2: Use Rotation Formulas

To find the new coordinates \((x', y')\), we use the rotation transformation formulas: \(x' = x\cos\theta + y\sin\theta\) and \(y' = -x\sin\theta + y\cos\theta\).

Step 3: Substitute Given Values

Substitute \(x = 0\), \(y = -2\), and \(\theta = 45^{\circ}\) into the formulas:1. \(x' = 0\cdot\cos 45^{\circ} + (-2)\cdot\sin 45^{\circ}\)2. \(y' = -0\cdot\sin 45^{\circ} + (-2)\cdot\cos 45^{\circ}\).

Step 4: Calculate Trigonometric Values

The values we need are \(\cos 45^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2}\).

Step 5: Calculate New Coordinates

After substituting the trigonometric values:1. \(x' = 0 + (-2)\cdot\frac{\sqrt{2}}{2} = -\sqrt{2}\)2. \(y' = 0 + (-2)\cdot\frac{\sqrt{2}}{2} = -\sqrt{2}\).Thus, the new coordinates are \((-\sqrt{2}, -\sqrt{2})\).

Key Concepts

Trigonometric Functions

Trigonometric functions play a crucial role in the rotation of axes, especially when dealing with coordinate transformations. These functions help us find relationships between angles and side lengths in a right triangle, which can be visualized in our task of rotating points on a coordinate plane.

When a point in the \(x\)-y coordinate system is rotated, we need to know the \( heta\) angle (the angle of rotation) to calculate the new position of the point. In this particular problem, \(\theta = 45^{\circ}\).

The two primary trigonometric functions used here are cosine (\(\cos\)) and sine (\(\sin\)). For \(\theta = 45^{\circ}\), the values are:

• \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\)
• \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\)

By substituting these values, we can determine the new coordinates after the rotation.

Coordinate Transformation

Coordinate transformation is the process of converting the coordinates of a point from one system to another. This is a key component in the rotation of axes. When we rotate the axes by an angle \(\theta\), every point in the \(x, y\) system needs to be recalculated to express their new position in the \(x', y'\) system.

The formulas for transforming the coordinates are:

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

Here, \(x\) and \(y\) are initial coordinates in the original system, and \(x', y'\) are their corresponding new coordinates in the rotated system.

In our example, \(x = 0\) and \(y = -2\). By applying the trigonometric values for \(\theta = 45^{\circ}\), we transform the point to \((x', y') = (-\sqrt{2}, -\sqrt{2})\). This process allows us to see how the same point shifts its position due to the rotation.

Angle of Rotation

The angle of rotation is fundamental in determining how a point's position changes in a coordinate plane. It tells us the degree to which we rotate the existing axes to find the new position in the transformed system.

An angle like \(\theta = 45^{\circ}\) specifies a counterclockwise turn. This angle is measured from the positive x-axis in the coordinate plane. Such rotations are common in computer graphics, robotics, and physics, where changes in orientation are needed.

Understanding the concept of the angle of rotation allows us to apply the appropriate trigonometric formulas. In our example, we used it to determine the exact shift from \(x, y\) to \(x', y'\). By leveraging the metrics of sine and cosine for \(45^{\circ}\), we computed the precise new location of the point, finally arriving at the coordinates \((-\sqrt{2}, -\sqrt{2})\).