Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

The graph of \(r=f(\theta)\) is rotated about the pole through an angle \(\phi .\) Show that the equation of the rotated graph is \(r=f(\theta-\phi)\).

Step-by-Step Solution

Answer

The equation of the graph \(r=f(\theta)\) rotated about the pole through an angle \(\phi\) is \(r=f(\theta-\phi)\).

Step 1: Define the coordinate transformation

By rotating the graph through an angle φ, we have created a new coordinate system \((\theta', r')\) such that \(\theta' = \theta - \phi\) and \(r'= r\). Our task is now to express the rotated graph \(r' = f'(\theta')\) in terms of the old coordinates.

Step 2: Substitute the transformation into the function

Now substitute \(\theta'\) with \(\theta - \phi\) into the function \(f'\). Doing this gives us:\(r' = f'(\theta') = f'(\theta - \phi)\).

Step 3: Simplify the function

Since in the original coordinate system we have \(r = f(\theta)\), the function \(f'\) must look the same as \(f\) but with the angles shifted by φ. Therefore, we simplify \(f'(\theta - \phi)\) as \(f(\theta - \phi)\) which gives us:\(r' = r = f(\theta - \phi)\). This is the equation for the shifted graph in terms of the old \(\theta\) coordinate.

Key Concepts

Coordinate Transformation

Coordinate transformation is a process of changing the coordinates from one system to another to simplify the equation or the computation. In the context of polar coordinates, such transformation becomes essential when we rotate the graph of a function. For instance, if you rotate a polar graph by an angle \(\phi\), the new position of each point can be expressed in terms of the original polar coordinates.

The original exercise involved rotating a polar graph, which implies a transformation where the new angle \(\theta'\) is related to the old angle \(\theta\) by the equation \(\theta' = \theta - \phi\). This is a simple subtraction, where \(\phi\) is the angle of rotation.

The key to understanding coordinate transformations is to recognize that although the position of the graph changes, the relationship between the coordinates remains consistent, just shifted by the transformation. Remember, the actual distance from the pole (the radius \(r\)) does not change with rotation in polar coordinates – it remains the same in the rotated system \(r' = r\).

Polar Graph

A polar graph represents data points in the polar coordinate system, where each point on the plane is determined by a distance from a reference point (the pole) and an angle from a reference direction. In the equation \(r=f(\theta)\), \(r\) represents the radius or distance from the pole, and \(\theta\) symbolizes the angle of the line from the pole to the point with respect to the positive x-axis.

A key feature of polar graphs is their ability to easily represent complex curves and shapes, such as spirals and circles, which might be difficult to describe with Cartesian coordinates. When dealing with function rotation in polar graphs, it's crucial to conceptualize the graph as fixed points on a piece of paper, with the paper being rotated. This helps when trying to visualize how rotating the graph affects the function equation and subsequently, the overall shape of the graph.

Function Rotation

Function rotation refers to rotating the graph of a function about a point. For polar coordinates, this point is typically the pole, or the origin. The exercise presented involves rotating the graph of \(r=f(\theta)\) about the pole through an angle \(\phi\).

To understand this concept, visualize a function as a shape on a graph being physically turned about a point. The rotated function's equation \(r=f(\theta-\phi)\) reflects this turn. The subtraction \(\theta-\phi\) indicates that you're effectively 'rewinding' each point's angle by \(\phi\), creating the image of the original graph, but rotated.

This idea of function rotation is an essential tool in advanced mathematics, where it aids in the analysis and graphing of complex functions, and in practical applications like computer graphics and navigation systems, where rotating reference frames is a common task.