Problem 6 - Precalculus: Mathematics for Calculus

Question

Determine the XY-coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(2,0), \quad \phi=15^{\circ}$$

Step-by-Step Solution

Answer

After rotation, the new coordinates are approximately (1.93, -0.52).

Step 1: Understanding the Problem

We are given a point with coordinates \((2, 0)\). We need to find the new coordinates \((x', y')\) after rotating the coordinate system by an angle \(\phi = 15^{\circ}\).

Step 2: Formula for Rotation

To find the new coordinates \((x', y')\) when the axes are rotated by an angle \(\phi\), we use the formulas:\[ x' = x \cos \phi + y \sin \phi \]\[ y' = -x \sin \phi + y \cos \phi \]

Step 3: Identifying Initial Values

From the given point, identify the initial coordinates as \(x = 2\) and \(y = 0\). The rotation angle is \(\phi = 15^{\circ}\).

Step 4: Calculate \(x'\)

Substitute the values into the formula for \(x'\):\[ x' = 2 \cos 15^{\circ} + 0 \sin 15^{\circ} \].Calculate \(\cos 15^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4}\). Thus, \(x' = 2 \times \frac{\sqrt{6} + \sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{2}\).

Step 5: Calculate \(y'\)

Substitute the values into the formula for \(y'\):\[ y' = -2 \sin 15^{\circ} + 0 \cos 15^{\circ} \].Calculate \(\sin 15^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4}\).Thus, \(y' = -2 \times \frac{\sqrt{6} - \sqrt{2}}{4} = \frac{-(\sqrt{6} - \sqrt{2})}{2} = \frac{-\sqrt{6} + \sqrt{2}}{2}\).

Step 6: Resulting Coordinates

The new coordinates of the point after rotation are:\[(x', y') = \left(\frac{\sqrt{6} + \sqrt{2}}{2}, \frac{-\sqrt{6} + \sqrt{2}}{2}\right)\]

Key Concepts

Coordinate Transformation

Transforming coordinates is an essential mathematical operation, especially when dealing with problems involving rotations or changing reference frames. In simpler terms, coordinate transformation enables us to look at a point from different perspectives. In coordinate geometry, this method is particularly useful when we want to discover how a grid or set of axes can change the descriptions of points. In our exercise, we took the original coordinates,

• (2, 0), and applied a specific transformation
• Our goal: find where this point lands in a new, rotated system.

Understanding coordinate transformations helps us easily switch between different systems or viewpoints. This can simplify complex geometry problems or make graphical interpretations more intuitive.

Rotation Angle

The angle of rotation is the key to understanding how a point's position changes in a new coordinate system. A rotation involves spinning the coordinate axes around the origin by a specific increment measured in degrees or radians. In our exercise, the rotation angle is 15 degrees. To perform this transformation:

• We use trigonometric functions such as sine and cosine to adjust the coordinates.
• This adjustment results in finding the new position of the point in the rotated system.

Rotating a system by different angles can drastically change how a point or entire figure is perceived or graphically represented. Thus, mastering this concept aids in advancing your understanding of geometric transformations.

Trigonometric Functions

Trigonometric functions like sine and cosine play a crucial role in rotation and coordinate transformation. They help determine the new positions of points after rotation. In a rotation, the initial coordinates are modified through the use of these functions. For example, in our routine task:

• The cosine function adjusts the x-coordinate.
• The sine function alters the y-coordinate accordingly.

This coupling ensures that every point's transition during rotation is mathematically precise. As seen in our problem, knowing the specific values of trigonometric functions for angles like 15 degrees is vital for accurate calculations. Therefore, getting accustomed to these fundamental calculations sharpens your problem-solving abilities in geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a field of geometry where points are defined using coordinates. This topic allows for a deeper comprehension of geometry using algebraic methods. In essence:

• It bridges the gap between algebra and geometry.
• It enables the precise description and analysis of geometric figures.

The use of coordinate geometry helps in representing geometric figures numerically, such as the translation of points and lines on a grid. By learning concepts like coordinate transformation through rotation using this method, one develops an extensive toolset for exploring and understanding multidimensional spaces. This knowledge is foundational for advanced studies into shapes, movements, and mathematical spaces.