Problem 74 - Precalculus with Limits

Question

Consider the graph of \(r=f(\sin\ \theta)\). (a) Show that if the graph is rotated counterclockwise \(\pi/2\) radians about the pole, the equation of the rotated graph is \(r=f(-\cos\ \theta)\). (b) Show that if the graph is rotated counterclockwise \(\pi\) radians about the pole, the equation of the rotated graph is \(r=f(-\sin\ \theta)\). (c) Show that if the graph is rotated counterclockwise \(3\pi/2\) radians about the pole, the equation of the rotated graph is \(r=f(\cos\ \theta)\).

Step-by-Step Solution

Answer

Upon rotating the polar equation \(r=f(\sin\ \theta)\) by \(\pi/2\), \(\pi\), and \(3\pi/2\) counterclockwise about the pole, the mapped polar equations respectively become \(r=f(-\cos\ \theta)\), \(r=f(-\sin\ \theta)\), and \(r=f(\cos\ \theta)\) respectively.

Step 1: Understanding rotation by \(\pi/2\) radians

Rotating a plot by \(\pi/2\) radians counterclockwise essentially means replacing \(\sin\ \theta\) by \(\cos\ \theta\). However, since the rotation is counterclockwise, we also change the sign, resulting in \(r=f(-\cos\ \theta)\).

Step 2: Understanding rotation by \(\pi\) radians

Rotating a plot by \(\pi\) radians counterclockwise means having to consider the opposite value of \(\sin\ \theta\). This is because rotating the graph by \(\pi\) radians is like reflecting the graph over the origin which means the sign reverses. Therefore, the equation of the graph becomes \(r=f(-\sin\ \theta)\).

Step 3: Understanding rotation by \(3\pi/2\) radians

When we rotate a plot by \(3\pi/2\) radians counterclockwise, we are replacing \(\sin\ \theta\) with \(\cos\ \theta\) and flipping the sign (since the rotation is counterclockwise). But because the rotation is for \(3\pi/2\) radians (which is \(\pi/2\) radians away from 2 complete loops), the sign is thus inverted again, resulting in the equation for the rotated graph being \(r=f(\cos\ \theta)\).

Key Concepts

Radian Measure

To understand the rotation of polar graphs, it's crucial to grasp the concept of a radian measure. A radian is a way of measuring angles based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

Since the circumference of a circle is given by the formula \(2\pi r\), where \(r\) is the radius, there are \(2\pi\) radians in a full circle. This means \(\pi/2\), \(\pi\), and \(3\pi/2\) radians correspond to quarter, half, and three-quarters of a full turn around a circle, respectively. These radian measures are commonly used when dealing with trigonometric functions and rotations in the polar coordinate system.

Polar Coordinates

Polar coordinates are an alternative to Cartesian coordinates for representing points in a plane. While Cartesian coordinates use a grid of vertical and horizontal lines, polar coordinates use a point called the pole (analogous to the origin in Cartesian coordinates) and a ray from the pole called the polar axis.

In polar coordinates, each point is determined by a radius \(r\) and an angle \(\theta\). The radius \(r\) is the distance from the pole, and \(\theta\) is the angle from the polar axis, measured in radians. This system is particularly useful for dealing with curves and shapes that are naturally circular or spiral, making it a fitting choice for analyzing rotations and the behavior of trigonometric functions.

Graph Transformation

Graph transformation refers to the manipulation of the visual representation of a function to reflect a mathematical change. Basic transformations include shifting (translating), reflecting, stretching, and rotating graphs.

When we discuss the rotation of a graph around the pole in polar coordinates, we are applying a transformation that moves points along circular paths centered at the pole. These rotational transformations follow the rules of trigonometry and are linked with the radian measurements of angles. For instance, rotating a graph by \(\pi/2\) radians represents a quarter turn counter-clockwise, altering the relationship between the radius and angle to produce a new equation for the graph.

Trigonometric Functions

Trigonometric functions are mathematical functions related to angles and sides of right-angled triangles, and by extension, to the circular motion. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), which relate an angle of a right triangle to the ratios of two of its sides.

In the context of polar graph rotation, sine and cosine functions are particularly relevant. These functions are periodic, and their values repeat every \(2\pi\) radians. As shown in the original problem's solutions, rotating the graph of \( r = f(\sin \theta) \) alters the trigonometric relationship because of the rotation's impact on the angle \(\theta\). Rotations can turn a sine-based function into a cosine-based one or change the function's sign, depending on the rotation angle, such as \(\pi/2\), \(\pi\), and \(3\pi/2\) radians, used in the rotations described.